_ o^^'

PJLU^^^X

i^y

^7A

©brarg

nf

(iIl|tB Dolumr ta tl|? gift of

3(1 jf/ffc

Digitized by tine Internet Arciiive
in 2010

littp://www.arcliive.org/details/firstcourseinstaOOjone

BELL'S MATHEMATICAL SERIES
ADVANCED SECTION

General Editor: WILLIAM P. MILNE, M.A., D.Sc
Professor or Mathematics, Univeksitv of Leeds

A FIRST COURSE IN STATISTICS

BELL'S MATHEMATICAL SERIES.

ADVANCED SECTION.

General Editor: WILLIAM P. MILNE, M.A., D.Sc,
Professor of Mathematics, University of Leeds.

AN ELEMENTARY TREATISE ON DIFFERENTIAL
EQUATIONS AND THEIR APPLICATIONS. By
H. T. H. PiAGGio, M.A., D.Sc, Professor of Mathe-
matics, University College, Nottingham. Demy 8vo.
Third Edition. I2s. net.

A FIRST COURSE IN NOMOGRAPIIV. By S.Brodetskv,
M.A., B.Sc, Ph.D., Professor of Applied Mathematics,
Leeds University. Demy 8vo. los. net.

AN INTRODUCTION TO THE STUDY OF VECTOR
ANALYSIS. By C. E. Weatherburn. M.A., D.Sc,
Professor of Mathematics, Canterbury University College,
Christchurch, N.Z. Demy 8vo. I2s. net.

ADVANCED VECTOR ANALYSIS, with Application to
Mathematical Physics. By the same Author. Demy Svo.
15s. net.

A FIRST COURSE IN STATISTICS. By D. Caradog
Jones, M.A., F..S.S. Second Edition, revised. Demy
Svo. 15s; net. Also in 2 parts. Part I., 7s. 6d: :
Part II., 9s.

THE ELEMENTS OF NON-EUCLIDEAN GEOMETRY.
By D. M. Y. SOMMERVILLE, M.A., D.Sc, Professor of
Mathematics, Victoria University College, Wellington,
N.Z. Crown Svo. 7s. 6d. net.

ANALYTICAL CONICS. By the same Author. Demy Svo.
15s. net.

LONDON : G. BELL AND SONS, LTD.

A FIRST COURSE

IN

STATISTICS

BY

D. CARADOG JONES, M.A., F.S.S.

LECTURER l^f SOCIAL STATISTICS AT LIVERPOOL UNIVERSITY

FORMERLY LECTURER IX MATHEMATIC

AT DURHAM UNIVERSITY

LONDON

G. BELL AIs'd SONS LTD.

1924

Fi7-st Published, . . 1921
Second Edition, revised, 1924

Printed in Great Britain by T. nnd A. Constable Ltd.
at the University I'ress, Edinburgh

PREFACE

Fifty years ago a large section of the general public were not
only uninterested in what we now call the social problem, but they
scarcely gave a thought to the existence of such a problem. They
felt vaguely perhaps, during periods of acute distress due to lack of
employment, that all was not well and they thought the Govern-
ment or possibly the big landowner was to blame, but only the
more enlightened realized the complexity of the body politic and
how fearfully and wonderfully it is made. To-day all this is changed,
and comparatively few imagine that a single panacea — the pro-
hibition of drink, the nationalization of land, or a levy on capital —
will cure all evils.

The very fact that nearly the whole civilized world has given
itself up for over four years to the destruction of life and the dragging
down of the social fabric in all countries on so vast a scale has
led to a surfeit and a reaction in which thoughtful men are eager
to take part in proclaiming again a common brotherhood and in
building a better world. Those who have always been interested
in this kind of architecture welcome the change of spirit, but they
also recognize the difficulty of the task undertaken and the need
for no little mental effort to second the good-will, which is the first
essential for success. To pull down no teacher is needed, but we
must learn to build.

This leads one to the subject of the present book. The man who
wishes his work to stand must make sure of its foundations. He
cannot afford to rest satisfied, as too often the politician and social
. worker do, with wild and ill-informed generalizations Avhere more
<i exact knowledge is possible, and there are few human problems in
the discussion of which some acquaintance with the proper treat-
ment of statistics is not in the highest degree necessary.

^
^

VI STATISTICS

Most people, however, are suspicious of figures. They imagine
that quantitative considerations must of necessity deaden all
feeling for the purely aesthetic or qualitative spirit which is the
very life of the phenomena observed or measured. But this surely
need not be the case. Kepler, when he succeeded in translating
the motions of the planets into the language of number was not, we
believe, the less but rather the more enamoured of the beauty and
order with which the whole of creation is clothed.

A second reason for suspicion is that partisans of one school or
another with more push than j)rinciple sometimes trade upon the
general ignorance of statistics to ' prove ' their own pet theories,
while others no less enthusiastic lead the credulous public into the
ditch, not witli malice intent, but because they are really blind
themselves to the right interpretation of the figures they so glibly
quote.

Although a concern in social questions led the present writer in
the first instance to study the theory of statistics, there is no reason
why this bias should prevent the book being of service to those who
■wish to know something of its application in other directions, seeing
that the general principles underlying the theory are the same in
all cases, and illustrations have been taken from any field, biological,
economic, medical, etc., just as the}^ suited the immediate purpose
in view.

The author makes no claim to any originality : he is no more
than a student seeking to put together, with some kind of system
and as he understands them, the simpler and more important ideas
he has gathered from other sources. The matter is entirely the
work of others, the manner only is his own, and he will be happy
to receive criticism if thereby he may learn more. His chief quali-
fication for writing is that he has had to worry through most of
his difficulties alone, and consequently he knows where another
student is likely to be in trouble better perhaps than the kind of
writer who is so quick as to be able to see through things at a glance
or, failing that, so fortunate as to be able to borrow immediate
light from others.

The book is divided into two parts. Practical]}^ all the first part
should be well within the understanding of the ordinary person.

PREFACE VU

Part II. is more mathematical, but an effort has been made through,
out to explain results in such a way that the reader shall gain a
general idea of the theory and be able to applj' it A\ithout needing
to master all the actual proofs. The whole is meant, not as an
exhaustive treatise, but merely as a first course introducing the
reader to more serious works, and, since real inspiration is to be
found noM'here so surely as at the source, it is intended to encourage
and fit him to pursue the subject further by consulting at least the
most important original papers referred to in the text, only enough
references being given to awaken curiosit}-. With the same inten-
tion a short chapter is inserted after the Appendix by way of sug-
gesting a few of the sources of statistics likely to be of interest to
the social student.

Some living WTiters, notably Professor Karl Pearson, have
contributed so largelj' to the development and application of
statistics that it is impossible to write upon the subject at all without
incorporating large parts of their ^^•or]v, and the least one can do
is gladly to record the benefit and pleasure one has received from
them. The author's indebtedness to the two most important
English text-books — Yule's Theory of Statistics and Bowley's
Elements of Statistics — will be evident also to any one who knows
these books, for they became so familiar through constant study
that he fears he may have drawn upon them unconsciously even
to the point of plagiarism in places.

Finally, he wishes specially to the acknowledge kindness of four
friends — Mr. Peter Fraser, Lecturer in Mathematics at Bristol Uni-
versity, without whose encouragement in the early stages the work
would never have been attempted ; Professor H. T. H. Piaggio,
University College, Nottingham, and Mr. A. W. Young, sometime
Lecturer at the Sir John Cass Technical Institute, London, whose
criticisms and suggestions were most valuable ; and Professor
W. P. Milne, of Leeds University, who, both as a practical teacher
and as Editor of this series, ungrudgingly gave his help and advice.

D. C. J.

NOTE TO THE SECOND EDITION

The kindly reception given to my book leads me to think that
it might appeal ■ to a wider circle of readers if they were not
frightened by the mathematical appearance of certain pages in the
second part. With this new issue it has been made possible to
obtain Part 1. and Part II. separately.

A selection of examples from London, B.Sc. (Econ.) papers has
been included by kind permission of the Authority concerned. It
is hoped that these may prove useful to students. An Index has
also been added. Otherwise no changes of importance have been
made in the text.

D. C. J.

September 1924.

CONTENTS

PART I

II.

III.

IV.

V.

VI.

VII.

VIII.
IX.

X.
XI,

Introdcctort — Earlt Historical Beginnincs : Logical
Development .....

Measurement, Variables, and Frequenct Distribution

Classification and Tabulation

Averages ......

Averages {continuiul) — Applications of Weighted Mean
Dispersion or Variability ....

Frequency Distribution : Examples to illustrate Calcu

LATING AND PLOTTING : SkEWNESS

Graphs — Correlation suggested by Graphical means

Graphs (continued) — Graphical Ideas as a Basis for Inter
polation : Reasoning made Clear with the Help of
Graphs or Curves .....

Correlation ......

Correlation — Examples ....

PART II

XII. Introduction to Probability and Sampling

XIII. Sampling {continued) — Formula for Probable Errors

XIV. Further Applications of Sampling Formulae
XV. Curve Fitting — Pearson's Generalized Probability

Curve .......

XVI. Curve Fitting {continued) — The Method of Moments fo

connecting Curve and Statistics .
XVII. Applications of Curve Fitting
XVIII. The Normal Curve of Error ....

XIX. Frequency Surface for Two Correlated Variables

APPENDIX

Certain Current Sources of Social Statistics

A Note on Tables to aid Calculation
MISCELLANEOUS EXAMPLES, Parts I and II .
INDEX

PART I
CHAPTER I

INTRODUCTORY

Early Historical Beginnings. Statistics, more or less valuable,
have been compiled in most civilized countries from very early
times. One reason for doing this on a large scale has been to
ascertain the man-power and material strength of the nation for
military or fiscal purposes, and we read in the Old Testament of
such censuses being taken in the case of the Jews, while among the
Romans also it was a common practice.

In England, as economic terms began to be used and their mean-
ings analysed, and especially during the period A^hen the mercantile
system prevailed, and the Government endeavoured so far as was
practicable to direct industry into channels such that it would add
most to the power of the realm, men tried frequently to base argu-
ments for social and political reform upon the results of figures
collected. A distinct advance had been made in the seventeenth
century when mortality tables were drawn up and discussed by Sir
William Petty and Halley, the famous astronomer, among others,
and their labours prepared the way for a more scientific treatment
of statistical methods, especially at the hands of one, Siissmilch, a
Prussian clergyman, who pubhshed an important Avork in 1761.

It is almost true to say, however, that until the time of the great
Belgian, Quetelet (1796-1874), no substantial theory of statistics
existed. The justice of this claim will be recognized when we
remark that it was he who really grasped the significance of one
of the fundamental principles — sometimes spoken of as the constancy
of great numbers — upon which the theory is based. A simple illus-
tration will explain the nature of this important idea : Imagine
100,000 Englishmen, all of the same age and living under the same
normal conditions — ruling out, that is, such abnormalities as are
occasioned by wars, famines, pestilence, etc. Let us divide these
men at random into ten groups, containing 10,000 each, and note
the age of every man when he dies. Quetelet's principle lays

A

2 STATISTICS

down that, although we cannot foretell how long any particular
individual will live, the ages at death of the 10,000 added together,
whichever group we consider, will be practically the same. De-
pending upon this fact insurance companies calculate the premiums
they must charge, by a process of averaging mortality results re-
corded in the past, and so they are able to carry on business without
serious risk of bankruptcy.

As a distinguished statistician once said, ' By the use of statistics
we obtain from milliards of facts the grand average of the world.'
But if the average resulting from our observations were subject to
violent fluctuation as we passed from one set of facts to another
cognate set there would be little satisfaction in fbiding it. It is
the comparative constancy of the average, if the number of our
observations is large enough, which makes it so important, as
Quetelet observed, for although the idea was not altogether new he
first reahzed how wide an application it had and how fruitful of
practical results it might prove.

Quetelet w^as born in Ghent, and taught mathematics in the
College there in his early youth. After graduating as Doctor of
Science he became Professor of Mathematics in Brussels Athenaeum
when only twenty-three years old, and later he was made Director
of the Brussels Observatory, in the foundation of which he had
taken a leading part. In 1841 he was appointed President of the
Central Commission of Statistics, where he was in a position to
render valuable assistance to the Belgian Government by his advice
on important social questions. He initiated the International
Statistical Congress, which has served to bring together the leading
statisticians of all countries, and the first meeting was held in 1853
at Brussels. His death occurred at the ripe age of seventy-eight.

Some idea of the extent of Quetelet's statistical researches may
be gathered from the titles of his chief works : (1) Sur Vhomme et
le developpement de ses facultes, ou essai de physique sociale (1835) ;
(2) Lettres . . . sur la theorie des probabilites appliquee aux sciences
morales et politiques (1846) ; (3) Du systeme social et des his qui le
regissent (1848) ; (4) L'Anthropometrie, ou mesure des differentes
facultes de Vhomme (1871).

In his writings he visuaHzes a man with quahties of average
measurement, physical and mental {Vhomme moyen), and shows
how all other men, in respect of any particular organ or character,
can be ranged about the mean or average man, just as in Physics
a number of observations of the same thing are ranged about
the mean of all the observations. Hence he concluded that the

INTRODUCTORY 3

methods of Probability, which are so effective in discussing errors
of observation, could be used also in Statistics, and that deviations
from the mean in both cases would be subject to the binomial law.

Hain in Vienna put some of Quetelet's ideas to good service in
1852, emplojdng a superior method for the calculation of statistical
variabiHty. ELnapp and Lexis in Germany, also following up
Quetelet's principles, made an exhaustive investigation several years
later of the statistics of mortaUty, and their work has been extended
in many directions, and in our OAvn time notably by Galton, Karl
Pearson, and Edge worth.

The name of Sir Francis Galton (1822-1911), to whose work as
a pioneer the science of Statistics owes so much, is deserving of
even greater honour than it has yet received. Founder of the School
of Eugenics, Galton himself came of famous stock, being grandson
of Erasmus Darwin and a cousin to Charles Darwin. He studied
medicine in early youth, but after graduating at Cambridge his
attention was turned to exploration, and the Royal Geographical
Society awarded him a gold medal on the results of his investiga-
tions in South- West Africa. His first great work on heredity was
not pubhshed till 1869, after he had already earned distinction in
other directions, for he was elected a Fellow of the Royal Society
in 1860. Ahve with new ideas, marvellously patient and persistent
in bringing them to the test of observation — quahties essential for
real scientific research — he set himseH to inquire into the laws
governing the transmission of characteristics, physical and mental,
from one generation to another. Large tracts of this ground have
since been carefully explored and mapped out by the school of
his great successor, Karl Pearson, who has originated formulae for
testing the extensive anthropometrical and biological data col-
lected. Largely as a result of their work it is now widely recognized
that ' the whole problem of evolution,' as Professor Pearson himself
has well said, ' is a problem in vital statistics — a problem of longevity,
of fertihty, of health, and of disease, and it is as impossible for the
evolutionist to proceed without statistics as it would be for the
Registrar-General to discuss the national mortality without an
enumeration of the population, a classification of deaths, and a
knowledge of statistical theory.'

Logical Development. The best way to approach the study of
any subject, if one had time, would be along the lines of its historical
development, but these lines seem so often to diverge from the
main theme, like branches from the parent stem of a tree, that

4 STATISTICS

when one tries to describe them the general effect is apt to be some-
what confusing. It is therefore usually the custom to adopt a
logical rather than a historical sequence, but it may assist the reader
to see the connection between the two and the miity which embraces
the whole if we now briefly trace the natural growth of the subject,
suggesting the steps we might expect it logically to take. This we
have tried to keep in view as nearly as possible in the succeeding
chapters, excejot that the order may have been altered here and
matter may have been omitted or inserted there as reason and
the elementary nature of the work dictated : —

1. Omng to the difficulty which the mind experiences in grasping
a large mass of figures, the necessity for an average arises to sum
up shortly the character of the mass, and various kinds of averages
are proposed.

2. An average proves insufficient alone to define the whole scheme
of observations, and other constants are invented to measure their
spread or dispersion about the average.

3. Considerations of space and the desire for some kind of system
lead further to the formation of tables with the observations classi-
fied in ordered groups.

4. The formation of these tables suggests the possibility of a
graphical representation of the numbers in the different groups to
bring out the nature of their distribution.

5. The impossibility of deaUng with a whole population results
in the selection of samples, and the comparison of one sample with
another introduces the subject of random errors.

6. The closer examination of this subject leads us into the domain
of mathematical probabiUty and discovers the probability curve, or
normal curve of error, first formulated in connection with the study
of errors of observation.

7. This same curve serves in the sequel to describe a certain
important type of statistical distribution, in which each observation
is determined by a multitude of so-caUed chance causes pulling this
way and that, so that it is impossible to foretell what the resultant
effect will be.

8. The failure of the normal curve to describe other common dis-
tributions, especially those which are unsjonmetrical in character,
leads to the development of skew varieties of curves which will
fit them.

9. The extent of connection between one set of data and a pos-
sibly related set is a natural subject for inquiry giving rise to the
theory of correlation.

CHAPTER II

MEASUREMENT, VARIABLES, AND FREQUENCY DISTRIBUTION

Measurement. There are two fundamental characteristics which
pertain to nearly all measurement : it is (1) relative : it involves
a comparison between one magnitude and another of the same kind,
and (2) approximate : the comparison in practice cannot be made
with absolute exactness.

A man's height, for example, is stated to be 5 ft. 8| in., but this
would convey little to one who did not know how long a foot was
and how long an inch was. The first step in the measurement is ,
made by comparing the man's length with a certain constant
length previously agreed upon as a standard or unit, namely, a
' foot ' ; he is placed to stand up against a scale which is divided
up into feet, and the highest point of his head is seen to come
somewhere betAveen the 5 ft. line and the 6 ft. line : he is there-
fore longer than five of these units, set end to end, but not so long
as six of them. To carry the measurement a stage further a smaller
unit has to be introduced ; each foot length of the scale is sub-
divided into twelve equal parts called inches, and the top of the
man's head is found to come somewhere between the 5 ft. 8 in.
line and the 5 ft. 9 in. line : he is therefore over 5 ft. 8 m., but
not quite 5 ft. 9 in. in height. For the next stage in the measure-
ment each inch of the scale has to be further subdivided into quarter-
inches, and the top of the man's head is found to come somewhere
between the 5 ft. 8 in. 3 qu. in. fine and the 5 ft. 9 in. line ; more-
over it is nearer, let us suppose, to the former line than to the latter.
In this case, then, we say that the man's height or length is 5 ft.
8| in., measured to the nearest quarter inch.

In measurement the decimal notation has very obvious advan-
tages, because each unit is always divided into ten equal parts to
get the next smaller unit. Thus a w^eight of 7 Idlogr. 5 hectogr.
3 decagr. 8 gr. 4 decigr. 3 centigr. can be expressed at once in
grammes, namely 7538-43 gr. ; hence if we were measuring to the
nearest decagramme, the result would be expressed as 754 decagr. ;
to the nearest decigramme, it would be 75384 decigr., etc.

6 STATISTICS

Similarly, a length of 12 kilom. 7 metres 2 centim, can be WTitten
12007-02 metres, or, in kilometres, 12-00702 kilom., or, to the nearest
decametre, 1201 decam., and so on.

The mere act of counting things of a like kind is, in a sense,
measurement of a primitive type, one thing being the unit, though
the measurement may in many such cases be exact ; for example,
we may count the number of persons in a room exactly. Even in
this type of case, however, the counting or measuring cannot
always be done accurately, but the inaccuracy arises from lack of
precision and miiformity in definition rather than from want of
power in the measuring instrument itself : e.g. in determining the
population of a city, inaccuracies may arise because of failure to
define exactly the boundaries of the city, or the time at which the
census is to be taken, or how to deal with the migration of the in-
habitants from or into the city, and with births and deaths during
the actual time of numbering.

Variables. By a variable is meant any organ or character which
is capable of variation or difference in size or kind. The diflference
may be measurable as in the case of head-length, height, tempera-
ture, etc., or not directly measurable as in the case of colour, intelli-
gence, occupation, etc . Further, the variation, when measurable, may
be continuous, or it may take place only by integral steps, omitting
intermediate values : population, for example, can never go up or down
by less than one, but if temperature is to change from 60 degrees to
61 degrees it must pass continuously through every intermediate
state of temperature between 60 degrees and 61 degrees.

In dealing with a measurable variable sometimes we are inter-
ested not so much m its actual value at a particular instant as in
the change which has taken place in its value during some specified
interval, but to gauge fairly the amount of this change it is necessary
to measure it relative to the original value of the variable. For
example, if we are told that the wages of a certain person have
gone up during the year to the extent of 3d. an hour, we cannot
say whether this is much or little to him until we know what his
wages Avere originally. The addition would be relatively much less
if he were a skilled patternmaker earning Is. 6d. an hour than it
would be if he were a chainmaker earning only 6d. an hour.* This
point can be met by stating, not simply the change in the value of
the variable, but the ratio of the new value to the old. For instance,
the patternmaker in the above instance has had his wages increased

[* Wages to-day are, ot course, much higher— the above figures are only hypothetical.]

MEASUREMENT, VARIABLES, FREQUENCY DISTRIBUTION 7

in the ratio of Is. 9d. to Is. 6d. It is important to notice that
this form of measurement is quite independent of the particular
vmits used ; if we take Id. as unit, the ratio=21/18=7/6, and if
we take Is. as unit, the ratio=l|/l|=7/6 just as before.

There are other ways of measuring this change in the value of
a variable. One of the commonest is to express it as a percentage
of the original value ; thus the patternmaker's increase is at the
rate of y^xlOO, or 16f per cent., which is simply the ratio of
increase in wage to previous wage multiphed by 100. The multiplier,
100, is quite an arbitrary factor, but it has obvious advantages : among
others, it works well with the decimal notation and it often serves
to put the result into a form which is greater than imity instead of
leaving it as a fraction. Again, a man who gets a dividend of £25
on an investment of £500 receives interest at the rate of yVo X 100,
or 5 per cent. ; in other words, this is the rate at which his capital
accumulates if the interest is added to it instead of being spent.

Annual birth rates and death rates, on the other hand, are best
expressed per thousand of the population, as estimated, say, at
the middle of the year in question ; e.g. the birth rate of the United
Kingdom in 1911 was 24-4 per thousand, and the death rate was
14-8 per thousand, which is equivalent to 244 and 148 per 10,000
of the population respectively. If we could assume the birth
and death rates to remain constant from year to year, and if we
could afford to leave migration out of account, the population
would be subject to exactly the same law of increase as capital
accumulating at compound interest [see Appendix, Note 1], thus : —

1. If P be the original population, and if the annual net increase
be at the rate of 25 per thousand, then

the population in 1 year's time=Px (1-025)

2 „ =Px (1-025)2

3 „ =Px (1-025)3
n „ =Px (1-025)".

2. If £P be the original capital, and if the annual increase be at
the rate of 2| per cent., then

the capital in 1 year's time=Px (1-025)

2 „ =Px (1-025)2

3 „ =Px (1-025)3
n „ =Px (1-025)".

Lest we may seem to have laboured to make plain what is really
a simple idea, it may be remarked that quite frequently confusion
arises with regard to percentage even in reputable quarters. As an

8

STATISTICS

illustration of the kind of mistake which, without thinking, is easily
made, the following argument has been taken from a monthly
circular sent out a httle while ago to the members of the Boiler-
makers' Society by their Secretary : Since July 1914, wages have
risen 15 per cent., the cost of living has gone up 45 per cent., therefore
the workers' real wages have fallen 30 per cent. This same argument
was quoted shortly after in one of the leading articles of The Man-
chester Guardian under the heading ' Prices and Wages,' and again
in The Labour Leader tersely as truth ' In a Nutshell,' but in
neither instance did it seem to have occurred to the writer that it
v/as inaccurate. It may be worth while for the sake of clearness to
show what the statement should have been : —

Wages.

Cost of
Living.

Ratio of Wages to
Cost of Living.

Same Ratio
multiplied by 100.

July 1914 .
October 1916 .

100
115

100
145

1

100

79

Since xil X 100 is roughly 79, this calculation shows that ' real
wages ' had fallen only about 21 per cent. (100—79=21), and not
30 per cent, as stated, between the two dates.

Index Numbers. A very important case of variables changing
with time appears in the discussion of changes in the value of
money as measured by the movement of prices of commodities,
introducing the notion of an index number. For example, supposing
the wholesale price of beef was 6d. a lb. at one date, 8d. a lb. at
another date, and 5|d. a lb. at a third date, the change might be
exhibited as in the following table : —

1st Date.

2ad Date.

3rd Date.

Price of beef

100

8cZ.
133

5|d.
92

Here 100, 133, and 92 are called index numbers, the price at the
first date being taken as a standard and denoted by 100, while
the prices at the other two dates are altered j)roportionally, so that

6:8:51=:: 100: 133:92.

Index numbers calculated on this principle have been published
systematically for several years by Mr. A. Sauerbeck (in the Journal

MEASUREMENT, VARIABLES, FREQUENCY DISTRIBUTION 9

oj the Royal Statistical Society up to January 1913, and continued
afterwards in The Statist under the supervision of Sir George Paish)
and in The Economist.

In Sauerbeck's index numbers the average wholesale prices of
forty-five commodities for the eleven years 1867-77 are taken as
the standard, being denoted each by 100 as above, and the prices
of the same commodities for any other year are then MTitten as
percentages of these standard prices. The commodities chosen are
various — food of all kinds (cereals, meat, potatoes, rice, butter,
sugar, coSee, tea), minerals (including coal), textiles, and sundries
(including hides, leather, tallow, palm oil, olive oil, linseed,
petroleum, soda, soda nitrate, indigo, timber). Articles of similar
character are grouped together ; naturally no class is exhaustive,
but the selection is a fairly representative one. A sort of general
average is then formed by combining all the results, and the move-
ment of this average is taken to measure changes in the value of
money. An example wiU make clear the way in which an index
number for each group and the general average are obtained.

The index number for each separate commodity may be first
calculated thus : —

Pkice of English Wheat.

Years.

QtX: Index Number.

1867-77
1912 .

s. d.

54 6 100

34 9 64

Now forming similar index numbers for each of the eight vegetable
and cereal foods and combining them together, we have : —

Index Numbers for Vegetable

AND

Cereal Foods

Years.

.2 c«

Mi

a> ^

3

o

S

_5'

o

6

'3

01

o

o
Cm

0)

bo O

o
O

>
<

1867-77 .

100

100

100

100

100

100

100

100

800

100

1912.

64

68

70

79

83

85

74

101

624

78

The figures m the last column but one are obtained by simply
adding the figures in the eight previous columns, and, dividing these

10

STATISTICS

results by eight, we get the average index number for the group
in 1912 as a percentage of that in the standard years 1867-77.

Treating all the other commodities in the same way we ultimately
get index numbers for all the different groups and for all com-
modities combined as follows : —

Index Numbers for different Groups and
FOR ALL Commodities.

No. of Commodities

8

7

4

19

7

8

11

45

Years.

m w

ri
o
o

"cS

£
"S
<1

UO ID

t«5C
o

o

a

i

.2 1

a \

a

o
O

1867-77

1912 ....

100

78

100
96

100
62

100
81

100
110

100
76

100

82

100
85

The index number for ' All Food ' is obtained by summing the
nineteen index numbers for the separate commodities which are
included in this class and dividing the result by 19. Similarly the
general index number for all commodities is obtained, not by
adding the numbers for the different groups and dividing by the
number of groups, but by adding the forty-five index numbers of
all the separate commodities and dividing the result by 45.

In The Economist the average prices of twenty-two commodities
for the years 1901-5 are taken as the standard, being denoted
each by 100, and the prices of the same commodities for any other
year are then written as percentages of these standard j)rices ; the
sum of these percentages is taken as the index number, and it is
a simple matter to divide by 22 if we wish to get the average per-
centage change. The following table explains the method of
calculation : —

Index Numbers representing Prices of Commodities

Date.

Cereals

and
Meat.

Ofciif Textiles.
Foods.

1

Minerals.

Miscel-
laneous.

Total.

Index No-

22.

1901-5 .

End of Dec. 1916

500
1294

300 500
553 1124-5

1

400
824-5

500
1112

2200

4908

100
223

MEASUREMENT, VARIABLES, FREQUENCY DISTRIBUTION 11

In this table five commodities are included under the head of
' Cereals and Meat,' three under ' Other Foods,' and so on. The
numbers in the last column are obtained by dividing those in the
previous column by 22.

It is clear that what is at bottom the same principle may be
applied in any case of a variable changing with time when we wish
to measure the extent of the change, so that the use of index numbers
is not confined to the problem of prices. We shall return again to
discuss one or two further points in connection with the same
subject in the Chapter on ' Averages.'

Frequency Distribution. So far we have been thinking more
particularly of the change which an individual variable, or a col-
lection of such variables, may undergo in the course of time, or the
difference between two values which the same variable may have
at two different instants of time, and how to measure it. Now
the science of Statistics is based upon the study of the crowd
rather than of the individual, although observations on individuals
have to be made before they can be combined together to produce
the crowd, just as individual income-tax schedules have to be
completed and combined before the balance-sheet of the State can
be drawn up. As we pass from one individual to another there
may be great differences in the organ or character observed — hence
the word variable already introduced — but in the mass these differ-
ences are merged together and lose their individual importance :
it is rather their resultant effect we seek to measure. In order
therefore to discover this effect it is necessary to make a collection
of individual observations and to analyse the results. Now if our
ultimate conclusions are to be safe the number of observations
must be considerable, and in order to be able to cope with them
and reduce them to some sort of system the first step in the analysis
consists in arranging them in different classes according to the
value of the variable under consideration.

It is to be noted that now we are dealing with changes in the
value of a variable as we pass from one individual to another at the
same period of time and mider the same general conditions, and not
with the change in a variable in the same individiial occurring with
the lapse of time. We wish, for example, to draw a distinction
between (1) the change in wages as we pass from one man to another
at the same time in the same trade, and (2) the change in wages of
the same man, or class of men, in the same trade occurring in a
given period of time ; in the first case we w ant to find the amount

12 STATISTICS

of diversity within the trade at some stated time, and in the second
our object is to discover whether an improvement has taken place
in the wages of a particular individual or a particular trade with
the passage of time.

In picturing variation of the first type the conception arises of a
frequency distribution where the observations are distributed in
ordered groups, with a number corresponding to each sho^^dng
how many, or how frequent, are the individuals possessing the type
of variable or character which defuies that group. More generally,
if a series of measurements or observations of a variable y are
made corresponding to a selected series of another variable x we
get a distribution, which becomes a frequency distribution when y
represents the frequency of events happening in a particular way,
or of individuals corresponding to a particular value of some
common variable or character, represented by x. Thus (1) the
boys in a school might be grouped according to their intelligence :
so many, dull ; so many, of ordinary intelligence ; and so many,
bright or above the ordinary. Again (2) in an inquiry into the
housing of the people in any town or district it would be necessary
to draw up a table showing the number or frequency of existing
tenements with one room, the frequency of tenements with two
rooms, the frequency of tenements with three rooms, and so on.
Once more (3) a zoologist, wishing to discover whether crabs of a
certain species caught in one locality differ in any remarkable way
from members of the same species caught in another locality, might
start by making measurements of the length of carapace or upper
shell for crabs of like sex in the two places and then proceed to
form frequency tables for each, setting out the frequency of crabs
for which the carapace length hes, say, between 5 and 6 millimetres,
the frequency with length between 6 and 7 millimetres, the frequency
with length between 7 and 8 millimetres, and so on. He would
then have in these tables some basis for comparing the specimens
caught in the two locaUties.

The three illustrations just used give three different types of
distribution corresponding to the three types of variable to which
attention has been drawn before. In the first, where the variable
or character observed is not measurable, doubt will sometimes
arise as to the appropriate class in which individuals should be
placed who seem to be on the border hne between dulness and
mediocrity or between mediocrity and brilliance, so that accurate
classification will greatly depend upon what is called the * personal
equation ' of the observer. The second illustration corresponds

MEASUREMENT, VARIABLES, FREQUENCY DISTRIBUTION 13

to the case where the variable changes not continuously but by
unit stages ; the choice of classes in such a case depends little
upon the observer unless the unit is very small compared to the
total range of variability ; for example, a tenement might either
definitely have two rooms or it might have three rooms, but it
clearly could not be put down as having 2 J rooms or 2g- rooms :
in other words, the only natural classification is so many tenements
with two rooms, so many wdth three rooms, so many with four
rooms, and so on, though here too some confusion might arise
through failure to define clearly Avhat is ' a room.' In the third
typo, where we can conceive of the continuous variation of the
character under observation, there would be nothing surprising in
the appearance of any value of the variable between the lowest
and highest values observed ; the choice of suitable limits for the
several groups becomes therefore in this case rather a delicate
matter which requires careful judgment.

We shall begin the next chapter with some general remarks
upon the subject of classification and tabulation.

CHAPTER III

CLASSIFICATION AND TABULATION

No part of Statistics is of more importance than that which deals
with classification and tabulation, and it is the one part for which
no very precise rules can be given. A neat arrangement of ideas
in the mind, capacity to express them clearly, and patience are
indispensable, but experience alone will convince one of the extreme
care which must be exercised if blunders are to be avoided and
time is to be saved in the long run. This has to be emphasized
because most people, until they have tried and failed, imagine
that to arrange things in classes and in tables is a straightforward
proceeding involving no great thought or trouble.

Abundant matter of a statistical character is published periodi-
cally in Blue-books, Government Reports, Reports of Local Authori-
ties, Directors of Education, Medical Officers of Health, Chief
Constables, Employers' Associations, Trade Unions, Co-operative
Societies, etc., but it needs a trained intelligence as a rule to assimi-
late it and turn it to further advantage. The larger the scale upon
which any inquiry is made, the more valuable should the results
be, granted that equal accuracy is possible on the large as on the
small scale, but it is fairly clear that mistakes of various kinds
have also much more chance of creeping into a large work than into
a small one. To appreciate the various and numerous possibilities
of error when the scope is wide it is enough to read the introduc-
tions to the Registrar-General's Reports on the Census from decade
to decade ; this should also impress the student with the care that
is necessary if he proposes to use such material for the investigation
of some other problem. It may seem a comparatively simple task
to abstract two sets of figures from a Census Report, to establish
a one-to-one correspondence between them, and to make deductions
therefrom, but such figures when taken from their context will
sometimes lead to absolutely unsafe, if not false, conclusions. The
exact meaning and limitations of any data can only be properly
appreciated by one who has been closely in touch with the persons
who have collected them, and it is therefore important, before

CLASSIFICATION AND TABULATION 15

attempting to re-classify or re-tabulate any old statistics for a new
purpose, to read very carefully through the notes made by the
original compilers.

Perhaps the best advice that can be given to any one in this
connection is that he should embark upon some small inquiry
which will necessitate the collection of statistics for himself ; the
final result of his efforts may seem disappointing, but the experi-
ence he will gain will be invaluable. Ideas for such an inquiry will
occur to him if he reads through some authoritative work on social
questions, e.g. Beveridge's Unemployment, the decennial Census
Reports, or The Miiiority Report on the Poor Law (1905). But he
must read with an open and critical mind, questioning particularly
the foundation for all statements as to cause and effect which may
be made. A few simple hints may be useful as to method of
procedure.

When he thinks he has discovered some subject of interest which
would appear to deserve examination, it will be well to put it
down on paper in order to get it clearly defined, because a precise
written statement is Hkely to carry one further than a shadowy
idea somewhere at the back of the mind which is hardly formu-
lated at all. When the actual collection of statistics is begun
it will almost certainly be found that it is impossible to solve the
original problem contemplated ; but that need not prevent further
progress — what is important is that the limitations should be
exactly realized, and this will be impossible unless the original
problem is clearly presented side by side Avith the nearest solution
obtainable.

The problem stated, the next thing is to set down categorically
a number of questions, the answers to which are to be the raw
material for the solution of the given problem. For the answers
let us assume the inquirer is dependent upon the goodwill of others,
either employers, or trade union secretaries, or public officials.
The questions in that case must be clearly, concisely, and courteously
phrased, and must not be capable of more than one interpretation.
In numbcK they should be few and in character not inquisitorial ;
moreover, the replies should be obtainable without any great labour
on the part of the persons approached. Here again it will be found
that the questions first set down are not all satisfactory : one will
be too vague ; another, though clear enough, may involve a con-
siderable search through a mass of other matter before it can be
properly answered ; while to another it might be impossible to give
an exact reply in any case. Revision and amendment may there-

16 STATISTICS

fore be necessary in the light of the first replies received, and the
inquirer will begin to see at this stage how far the solution to his
original problem is really possible.

Wlien the bulk of the returns have come in they should be critically
examined one by one. A number will, for one reason or another,
be worthless, and they must be discarded ; as for the remainder,
if the questions were well chosen, the answers should not be difficult
to interpret and classify ; the most successful questions are those
to which a simple ' yes ' or ' no ' in reply gives all the information
required ; numerical answers are less easy to deal with, especially
if there is the least chance of misunderstanding on either side as
there often is, for example, in the case of observations which are
on the border line between two classes.

Tables should then be dra^vn up and the headings to the dinerent
columns of the tables should state concisely and exactly what the
figures below represent. So far as possible any one should be able
readily to grasp their general meaning without being obliged to
wade through a page or two of written explanation ; if any heading
cannot be clearly expressed in a few words it may be helped out
by a further note at the bottom of the page, but too many such
notes are to be avoided.

Finally, a summary should be made of the various conclusions
suggested by a study of the tables. Some of the points raised in
the course of the inquiry will perhaps be only incidental to the
main problem under discussion, but may still deserve a passing
reference. It will also be of advantage to follow up the summary
by any recommendations which can be fairly based on the con-
clusions obtained, when the problem is such that recommendations
are expedient, and, if ultimately the whole is of sufficient value to
be printed, emphasis can be introduced where necessary by suitable
variations in type.

For this part of the work considerable judgment is necessary
which can only be acquired by long training — a faculty to pick out
the real from the false and an eye to distinguish the important from
the trivial. A sense of numerical proportion too is desirable inci-
dentally ; one of our leading exponents on finance in a book deaHng
with the meaning of money uses a very interesting illustration which
is perhaps worth quoting here to show how even an acute mind
may on occasion prove itself curiously lacldng in such a sense.
He is seeking to show how the credit system of the country is built
upon a foundation composed of a little gold and a lot of paper ;
for this purpose he amalgamates together the balance-sheets of half

CLASSIFICATION AND TABULATION

17

a dozen big banks, and ijrovcs that their liabili,ties on current and
deposit account amounted at a certain date prior to 1914 to 249
million pounds, while the cash in hand and at the Bank of England
was 43 millions. Of the 43 millions he estimates that roughly
20 millions would be cash in the Bank of England, and further
that about two-thirds of this 20 millions would be represented really
by securities and not by gold. Hence he concludes that to support
this vast erection of credit there would only be £6,666,666 of actual
gold. Thus after talking throughout in millions the author closes
by giving his result true apparently to a pound !

Much may be learnt as to methods of classification and the
drawing up of tables by a careful study of those which appear in
various official reports, and a few such tables are reproduced in
the pages which follow.

Table (1). Condition as to Cleanliness of
School Children in Surrey.

Cleanliness.

5 years, 1908-12. 79,070 children inspected.

Above the average .
Average
Below average
Much below average

15-4 per cent.
76-5

7-6

0-5

Table (2). Condition as to Infectious Diseases op
School Children at Different Ages in Surrey (1913).

Age Groups inspected

5-6

8-9

13-14

Total at
All Ages.

Numbers inspected

5,191

5,151

4,902

15,304

Proportion who before inspec-

tion had suflEered from —

per cent.

per cent.

per cent.

per cent.

Diphtheria .

1-3

3-5

5-4

3-4

Scarlet fever

2-7

7-2

10-9

6-9

Measles

55-3

79-3

84-6

72-9

Whooping cough

41-8

56-4

54-3

50-9

German measles

2-9

51

7-5

51

Chicken pox

261

401

38-6

34-9

Mumps

10-6

220

29-8

20-7

No infectious diseases .

18-9

61

4-7 1

100

No definite information

3-3

2-2

0-9

2-2

18

STATISTICS

Table (3). Height of School Children according to
District, Age, and Sex (1913).

Age
Groups.

Boys.

Girls.

Nos.
measured.

Average

Height

in inches.

Average
in c

Surrej'.

Height
ms.

England

and
"Wales.

Nos.
measured.

Average

Height

in inches.

Average Height
in cms.

England
Surrej-. and
Wales.

5-6

8-9

13-14

2724

2578
2529

41-4

47-8
57-0

105-2
121-4
144-8

103-4
120-4
142-4

2467
2573
2433

41-3
47-5
57-9

104-9
120-7
147-1

102-6
119-4
144-2

The first four are taken from the Annvxil Report of the School
Medical Officer for the County of Surrey, 1913. The first is an
example of single tabulation showing the distribution according to
cleanliness of children inspected in the elementary schools. The
second is an example of double tabulation, showing the distribu-
tion according to age of school children who at some period before
the date of inspection had suffered from certain infectious diseases.
The third is an example of quadruple tabulation, showing the dis-
tribution of school children according to height, district, sex, and
age. Thus in the first case we have one factor brought into reUef,
viz. cleanliness ; in the second case we have two factors, age and
disease ; in the third case we have four factors, height, district,
sex, and age.

When we have two or more factors tabulated together as in cases
(2) and (3), we may be sometimes led to discover a connection of
some kind, possibly causal, between them, and the search for such
a connection, or correlation as it is called, represents one very useful
purpose to Avhich tabulation may be jjut. Table (4) is an illustra-
tion of this. It is the result of certain measurements carried out in
order to discover the effect of employment out of school hours upon
the physical concHtion of boys. The particular factor examined as
the possible cause of evil in this comiection is lack of sleep, and
the figures given certainly seem to warrant a closer examination
into the matter.

CLASSIFICATION AND TABULATION

19

Table (4). Physical Condition of certain Boys according
TO Hours of Sleep Obtained.

No. of Hours
Sleep obtained.

No. of Boys
examined.

Average

Height in

inches.

Average

"Weight in

lbs.

Nutrition.

Percentage

above

average.

Percentage
average.

Percentage

below

average.

7 to 8 .

8 to 9 .

9 to 10 .

10 to 11 .

11 to 12 .

14

80
296
280

50

54-5
55-4
56-4
57-9
59.0

71-3
73-9
79-3

83-2
87-0

71
101
15-3

22-8
220

35-8
65-9
64-5
66-5
68-0

57-1
240
20-2
10-7
100

Tables (5) and (6) are two illustrations of neat tables, containing
a large amount of information in a small space, set out in such a
form that the eye can easily take it in— and that is the main purpose
of tabulation. These examples are selected from the Sixteenth
Abstract of Labour Statistics of the United Kingdom, Cd. 7131.

In Table (6) note the classification of age groups : it is not ' 5 to
10 years,' ' 10 to 15 years,' and so on, but ' 5 and under 10 years,'
' 10 and under 15 years,' and so on. This removes difficulties at
the border lines between two classes ; the difficulties are not com-
pletely removed, however, unless there is some understanding as
to what shall constitute under any particular age. Shall it be six
months under, or one day under, or one hour under ? This sort
of ambiguity has more importance in some cases than in others.
Suppose, for example, we were classifying men according to their
height : a group of the typo ' 60 inches and under 62 inches,'
assuming that measurements were made to the nearest half-inch,
would really include all men who were ' 59f inches and under
61 1 inches ' ; because one who measured anything from 59| in.
to 60 J in., being nearer to 60 in. than to 59| in. measuring to
the nearest half-inch, would be registered as 60 in. in height, while
one who measured anything from 61| in. to 62J in., being nearer
to 62 in. than to 61 1 in., would be registered as 62 in. in height.

Another point to be noted is that in general people making
returns seem to have a psychological weakness for round figures,
so that a man in the neighbourhood of 40 years of age, for example,
is apt to record himself as actually 40 although he may really

^0

STATISTICS

Table (5). Classification of Overcrowded Tenements—*
England and Wales (1911).

Urban Districts.

Rural Districts.

Total.

Occupauts
thereof.

Occupants
thereof.

Occupants
thereof.

Tenements

WITH

No. of
Over-
crowded
Tene-
ments.

No. of
Over-
crowded
Tene-
ments.

No. of
Over-
crowded
Tene-
ments.

No.

Per-
cent-
age of
total

No.

Per-
cent-
age of

total

No.

Per-
cent-
age of
total

popu-
lation.

0-7
2-5
3-0
2-2

popu-
lation.

popu-
lation.

0-6
2-2
2-8
2-2

1 room .

2 rooms .

3 rooms .

4 rooms .

56,290
119,69.5
107,892

64,470

206,022
712,613
847,937
624,747

1,545
15,397
22,380
17,341

5,748

91,458

175,988

167,969

0-1
1-2
2-2
2-1

57,835
135,092
130,272

81,811

211,770

804,071

1,023,925

792,716

5 or more
rooms .

1

21,200

251,405

0-9

4,700

55,585

0-7

25,900

306,990

0-8

Table (6). Population grouped according to Age —
England and Wales (1911).

males.

Age Groups.

Uruan Districts.

Rural Districts.

All Districts.

Number.

Percentage.

Number.

Percentage.

Number.

Percentage.

Under 5 years
5 and under 10 vears
10 „ 15 „
15 „ 20 „
20 ,, 30 „
30 „ 40 ,,
40 ,, 50 „
50 „ 60 „
60 „ 70 „
70 and upwards

1,517,432
1,431,900
1,341,586
1,267,500
2,332,135
2,094,934
1,-556,818
1,042,868
612,741
296,240

11-3]
10-6 1 ^ „

9-9 r ^^

9-4 )
17-3^
1.5-5 144-4
ll-OJ

7-71

4-5 U4-4

2-2j

418,681
415,395
400,045
387,395
626,300
542,370
444,360
333,308
2.30,306
147,228

10-6^

10-3 P^ ^
9-8J
15-9^
13-7 HO-9

ii-sj

8-4^
. 5-8yi7-9
3-7J

1,936,113
1,847.295
1,747,631
1,654,895
2,958,4.35
2,637,304
2,001,178
1,. 376. 236
843,047
443,474

ii-n

10-0 Li .9
10-0 1 ^^ -

9-5j
17-0^
15-1 U3-6
ll-oj

7-9-1

4-8 } 15-2

2-5 j

Total

13,494,100 1000

3,951,448

100-0

17,445,608

100-0

* For the purpose of the Census Reports 'ordinary tenements which have more
than two occupants per room, bedrooms and sitting-rooms included,' are considered
overcrowded.]

CLASSIFICATION AND TABULATION 21

be 39 or 41 years old. To diminish the error arising from this fact
it is usual, when not otherwise inconvenient, to fix the centres
of the class-intervals at round figures : e.g. to take ' 15 and under
25 years,' ' 25 and under 35 years,' etc., in preference to * 20 and
under 30 years,' ' 30 and under 40 years,' etc. Where there is
any known bias in the data, as, for instance, in the familiar case
of certain women who consistently register themselves as younger
than they really are, a correction can be made in the final figures.

In any frequency distribution where we wish to group a number
of observations according to the magnitude of some common
variable, as in Table (6) a number of males grouped according to
age, the question arises — ' How many grouf)s should there be ? '
With tliis question is involved also the size of the corresponding
class-interval, and this should be so large that, with possible excep-
tions at either extremity of the table, there are a fair proportion of
observations to each class or group ; and, contrariwise, it should
be so small that all the observations in any one group may be
treated practically as if they were located at the centre of the group
so far as the variable in question is concerned, e.g. it should be
possible to treat males recorded in class ' 50 and under 60 years,'
where the interval is 10 3^ears, as if they were all of age 55 years. It
will be found in general that a number of groups somewhere in the
neighbourhood of 20 is the most satisfactory, granted that the
number of observations is reasonably large, although in some cases
it is impossible to split up the unit of class-interval, and we are
obHged to be satisfied wdth a smaller number of groups on this
account : Table (5) is a case in point where we are tied down to
one room as the class-interval. In Table (6) the class-interval
varies, being only 5 years at first, and afterwards 10 years, but
as a rule the labour of calculation of the different statistical constants
we require is considerably simplified if it is possible to keej) the
size of the class-interval the same for each group.

CHAPTER IV

AVERAGES

Common Average or Arithmetic Mean. Let us consider one of the
commonest meanings of the term average. If a train travels a
distance of 180 miles in 3 hours we say that it has been moving
at 60 miles an hour. By this we do not mean that its speed is
always 60 m/h, never more, never less, but that if it had moved
always at that uniform speed it would have accomplished its
journey in exactly the same time. As a matter of fact, during
some instants it may have been moving at a much slower rate
than 60 m/h, but, if so, it must have made up for this slackness
by travelling at a much faster rate than 60 m/h during other
instants, so that on the whole a balance was effected, and, as we
say, the speed averaged out at 60 m/h.

Again, suppose the wages of three men are : A, 27s. a week ;
B, ISs. a week ; C, 30s. a week. We should say that the average
wage of the three was equivalent to

|(27+18+30)s. = 2os. a week.

In other words, if A, B, and C were all under the same employer,
and if, instead of paying them different amounts, he wanted to
pay them all equally, he would have to give each man 25s. a week,
assuming that his total wages biU was to remain unaltered. This
method of measurement gives what is kno\\Ti as the arithmetic
mean, or, more simply, the mean.

Once more, in discussing the state of the labour market as regards
different trades, when we wish to compare one with another, it is
not the actual numbers unemployed in each trade that are quoted,
but these numbers expressed as percentages of the total numbers
employable in each trade.

In each of these three cases we reduce our observations or
measurements to a sort of common denominator, so that they may be
mentally compared or contrasted more readily with other observa-
tions of a similar character. Thus we have in mind a certain mean

22

AVERAGES 23

train speed per hour, or mean wage per week, or mean percentage
out of work, as the case may be.

An average then in general we may regard as one of a class
of statistical constants (others of which we are to meet later) which
concisely label a set of observations or measurements pertaining
to a common family. It is designed to describe the family type
more nearly than is possible by observing any chance member, and in
value it should therefore come somewhere near the middle of the
family group, so that if the individual members of the family
chance to be equal each to each in respect to the organ or character
observed it should have the same value as they have. This consti-
tutes a test for the validity of any fornmla giving the average of a
set of observations : e.g. we might, if we wish, define the average
of three numbers, p, q, r to be, not \{'p-\-q-\-r) but

for (1) this formula, too, can be shoA\Ti to give a number intermediate
in value between the greatest and least of the numbers p, q, r ;
also (2) if we put p=q=r=k (say), the formula reduces to

Clearly the range of choice for the definition of an average is
infinite, though only a few definitions give averages which have
proved their utility and come into general use. Of these the most
important is the common mean already introduced, with its ex-
tension, the weighted mean, but at least two others deserve special
consideration, the median and the mode.

Median. In any observed distribution if all the individuals
can be arranged in order of magnitude of the character or organ
observed, which may be conveniently done when they are not very
numerous, the median organ or character will be that pertaining to
the individual half-way along the series, so that there are in general
an equal number of individuals above and below the median.
For instance, if seven boys of different heights be placed to stand in
a row, the tallest first, the next tallest next, and so on, the median
height is the height of the fourth boy from either end. If there
are an even number of boys, say eight, it would be natural to take
as median the height midway between that of the fourth and that
of the fifth boy.

When the items are numerous they are frequently grouped into
classes, as we have seen, such that all in the same class are reckoned

24 STATISTICS

to have some value lying between the extreme limits of that class.
We should then, as before, halve the total number of observations
to fix the particular individual which defines the median organ or
character. This would enable us to pick out the group in which
the median lies, and on reference to the original record of observa-
tions, assuming it was at hand, it would be a simple matter to
identify the median.

If the original record be not available, however, it will be neces-
sary to jDroceed to get the best value we can for the median in some
other way. Consider, for example. Table (7), showing the distribu-
tion of marks obtained by 514 candidates in a certain examination.
We begin by rearranging the data in the manner shown below
Table (7). Now in accordance with the definition the median in
marks should, strictly speaking, be midway between the marks
assigned to the 257th candidate and the marks assigned to the
258th candidate : in fact, the marks corresponding to candidate
number 257-5, if it were possible for such a candidate to exist.
But we are ignorant so far as Table (7) goes of the marks gained
by either the 257th or the 258th candidate, though it is possible,
by the simple proportional process known as ' interpolation,' to
calculate approximately the marks we require. We think of all
the candidates as forming an ordered sequence, ranged one after
the other according to their marks just like the boys of different
heights, and the table shows that in this mental picture

the 231st candidate gets approximately 30 marks, while
„ 318th „ „ „ 35 „

Hence candidate number 257-5, if one existed, ought to get a
number of marks somewhere between 30 and 35. But, in tliis
neighbourhood of the sequence,

a difference of (318-231) candidates corresponds to a difference

of 5 marks, therefore
a difference of (257-5-231) candidates corresponds to a difference

of (.^Vx 26-5) marks. '^

Thus the marks obtained by candidate number 257'5 are ap-
proximately = 30+ -ir X 26-5

=31-523,

and this may be taken as the median.

On examining the actual marks-sheet it v/as found that 252
candidates obtained 31 marks or less, and 273 candidates obtained

/•«.

AVERAGES

25

32 marks or less, so that the real median was 32, because this was
the number of marks gained by both the 257 th and the 258th
candidates. The number 31-523 found above, however, would be
a good approximation to take for the median when all the informa-
tion at our disposal was that showTi in Table (7).

Table (7). Makks obtained by 514 Candidates in a
CERTAIN Examination.

Marks Obtained.

No. of
Candidates.

Marks Obtained.

No. of
Candidates.

1 to5
6 to 10
11 to 15
16 to 20
21 to 25
26 to 30
31 to 35

5
9

28
49
58

82 '
87

36 to 40
41 to 45
46 to 50
51 to 55
56 to 60
61 to 65

79
50
37
21
6
3

Total

514

The table is to be read as follows : —

5 candidates obtained 1, 2, 3, 4, or 5 marks,

9

6, 7, 8, 9, or 10

and so on.

By straightforward addition it can evidently be rearranged so
as to read thus : —

5 candidates obtained not more than 5 marks.

14

42
91
149
231
318
397
447
484
505
511
514

10
15
20
25
30
35
40
45
50
55
60
65

26 STATISTICS

It will be noted that in calculating the median no use is made of
the marks of any of the candidates except those in the two groups
in the immediate neighbourhood of the median, and it is one of
the great advantages of this average that it can be found when an
exact knowledge of the characters of the more extreme individuals
in the series is not in our possession, and even wtien their measure-
ment is impossible : it is enough if they can be roughly located.
The arithmetic mean on the other hand is often unduly influenced
by abnormal individuals which are not really typical of the popula-
tion in which they appear.

Mode. If we measure or observe some organ or character for
each individual in a given population, the mode, as its name sug-
gests, is simply the organ or character of most fashionable or most
frequent size. A large draper, for example, will have collars of
several different shajies and sizes in his shop, but the fashionable
shape and the predominant size correspond to the mode : it is the
mode that sells most readily, and the intelligent draper will always
have it in stock. Again, in Table (2), the disease mode or fashion-
able disease among certain school children inspected in Surrey in
1913 was measles, for a greater percentage of children had suffered
from measles than from any other of the diseases recorded.

Now when the variable in which we are interested is ' discrete,'
that is, when it changes by unit steps, leading to classes like ' tene-
ments with 1 room,' ' tenements with 2 rooms,' ' tenements with
3 rooms,' and so on, it is an easy matter to pick out the class of
greatest frequency : thus, in Table (5) there are more overcrowded
tenements with 2 rooms than with any other number of rooms
in the urban districts, so that 2 is the mode so far as this character
(number of rooms) is concerned, whereas in the rural districts 3 is
the mode, for there are more overcrowded tenements with 3 rooms
than with any other number. There may be ambiguity, however,
in determining the mode in this way for a grouped frequency dis-
tribution when we are dealing with an organ or character subject
to ' continuous variation.' To cover such cases the modal value
has been defined as that value for which the frequency per unit
variation of the organ or character is a maximum. The precise
significance of this Avording will only be appreciated after discussmg
frequency curves : at present it must suffice to give a practical
illustration of how the ambiguity arises and calls for some more
refined treatment.

For this purpose turn again to the examination marks in Table (7),

AVERAGES

27

from which it appears that the mode, if it is to be the marks obtained
by the greatest number of candidates, should he in the group
(31 to 35), since there are 87 candidates ^vith marks between these
Hmits, and this number exceeds that in any other group. But
how are we to decide the exact point in the interval (31 to 35) which
is to correspond to the mode ? Shall it be 33 ? We might say
' yes ' if the distribution were perfectly symmetrical on either side
of the (31 to 35) group, but if we examine the neighbouring groups
we see that the balance leans rather more heavily to the (26 to 30)
group with a frequency of 82 than to the (36 to 40) group with a
frequency of 79, and we might allow for this by interpolating in
some way— ignoring, of course, any errors which may occur in the
frequencies themselves owing to the observations being generally
limited in number. But the pull in the direction of lower marks
becomes still more pronounced to our minds when we contrast
also the frequencies in the next groups on either side, namely
58 and 50. So we might go on until the influence of the whole
field of observations comes into action.

Now it so happened that in this particular case the original
marks-sheet was to be seen, and a regrouping of the candidates as
in Table (8) makes it clear that the value found in this way for the
mode may be artificially displaced sometimes to a serious extent
by the particular method of grouping adopted. Thus, according
to this new arrangement, the mode would seem to lie in the interval
(28 to 32), the mid-value of which differs materially from 33, the
mid- value of the previous maximum frequency group.

Table (8). Marks obtained by 514 Candidates in a
CERTAIN Examination (Alternative Grouping).

Marks Obtained.

No. of
Candidates.

Marks Obtained.

No. of
Candidates.

3 to 7
8 to 12
13 to 17
18 to 22
23 to 27
28 to 32
33 to 37

10
17

35
56

47

108

74

38 to 42
43 to 47
48 to 52
53 to 57
58 to 62
63 to 67

73
45
31
12
3
3

Total

514

28 STATISTICS

[It should be observed that while an alteration of the grouping
may also affect the median, it does not affect it nearly to the same
extent : e.g. the median determined from Table (8) is 31-3, which
differs little from 31-5 the value obtained by the first grouping.]

If, again, we combine the results of our two groupings to find
the mode we might be tempted to conclude that it lies somewhere
between the limits 31 and 32, but on examining the original records
it was discovered that the real mode was 28. The frequency
distribution of candidates in this neighbourhood was in fact very
interesting ; it ran as follows : —

Number of candidates who obtained 25 marks=14

26 „ -=10

27 ,

, = 6

28

, =33

29 ,

, =17

30 ,

, =16

The explanation of this peculiar distribution seemed to be that
28 marks were required for a candidate to pass, and apparently as
many candidates as possible were pushed over the pass line : if,
on the first marking, a candidate was found to want only one mark
to pass, the examiner presumably looked through his paper again
and did his best to find an answer which by kindly treatment
might be granted an extra mark. The effect of this leniency was
ultimately to leave only 6 candidates in the division immediately
below the pass line, and to swell the number immediately above
to 33, which thus made 28 easUy the ' most fashionable ' mark of
any, the next largest group of candidates being only 21. It will
be observed that even a candidate who wanted 2 marks to pass
was treated in the same tolerant fashion, although it is not so
easy, of course, for a conscientious examiner to discover two extra
marks as it is to discover one ; and if the candidate is 3 marks
below the pass line it is still harder to give him the necessary lift
to carry him over. Thus in the final list we fimd more condidates
with 26 marks than with 27, and stUl more with 25 than with 26.
If the above diagnosis is correct, and all marks-sheets tell the same
tale, who shall again say that examiners do not temper justice with
mercy ?

This example has illustrated fairly clearly the difficulty of fixing
the mode with any great precision by mere inspection when the
individuals are arranged in groups, the value of the variable under
discussion lying between prescribed limits for each group. While

AVERAGES 29

it is possible to get a rough approximation to its value in this way,
we conclude that for a really satisfactory determination we require
some method which makes use of the whole distribution, as in the
determination of the mean, and not merely of the portion in the
supposed neighbourhood of the mode. This must be left to a later
chapter ; we shall only point out before passing on that there
may sometimes be more than one mode in a given frequency dis-
tribution just as there may be more than one fashionable type of
collar which it is expedient for the draper to stock in large quan-
tities. The second grouping in the examination example suggests
such a possibilitj'', for it will be noticed that the frequencies of
candidates do not rise steadily to a single maximum at 108 for
class (28 to 32), and then fall steadily : there is a previous rise and
fall in the neighbourhood of class (18 to 22).

Weighted Mean. Let us suppose a farmer employs for the
harvest 5 men, 3 women, and 4 bo3'S. In estimating the amount
of work they can do in a given time it is clear that in general a
woman or boy cannot be reckoned as equal to a man. He must
therefore decide what ' weight ' must be given to each in proportion
to a man. If a woman's work be taken, for example, to be three-
quarters as effective and a boy's work to be half as effective as
that of a man, we have as the appropriate proportional weights

1 : f : 1 or 4 : 3 : 2.

Hence 5 men, 3 women, and 4 boys would on the average be equiva-
lent in output to

(5+3xf+4x^) men

4x5+3x3+2x4
= men

=9i men.

An average of this type is called a weighted mean, 1, f, and
I being the weights, because they tell us what weight to give to
each separate worker in calculating the average.

Let us consider the effect such weighting has in general upon a
mean, and for this purpose we shall test it on a set of index numbers
measuring rents in certain groups of to^Tis in 1912, as given in a
Report on the Cost of Living of the Working Classes issued by the
Board of Trade (Cd. 6955).

80

STATISTICS

Table (9). Mean Index Numbers of Rents for certain

Geographical Groups of Towns in 1912 (with reference

TO Middle Zone of London as standard — 100).

(1)

(2)

(3)

(4)

(5)

(6)

Geographical Group.

Rents.

No. of
Towns
included
in the
Group.

Each

Group

counting

as 1.

Arbitrary

Weights.

Approxi-
mate sub-
multiples
of Nos. in
previous
column.

Northern Counties and Cleve-

land ....

660

9

27

3

Yorkshire (except Cleveland)
Lancashire and Cheshire

58-5
56-9

10
17

54
45

I

Midlands ....

52-3

14

125

14

Eastern and East Midland Cos.

53-4

7

63

7

Southern Counties

63-7

10

14

2

Wales and Monmouth .

64-8

4

22

2

Scotland ....

62-0

10

178

20

Ireland ....

51-7

6

55

6

Average

••

58-4

58-8

57-6

57-6

The first mean in the above table, 58-4, is obtained by multipl}^-
ing (or weighting) the mean rent of each geographical group by the
number of towns in the group, given in col. (3), adding the numbers
so obtained, and dividing the total by the total number of towns,
thus : —

9(66-0)+ 10(58-5)+ • • • +6(51-7)

9 + 10 +

+ 6

This is simply the arithmetic mean treating each town as unit.

The second mean, 58-8, is obtained by adding the mean rents of
all the groups and dividing by the total number of groups, thus : —

66-0+58-5-

+51-7

1 + 1 +

+ 1

This is the arithmetic mean treating each geographical group as
unit.

The third mean, 57-6, is obtained by multipl3dng, or weighting,
the mean rent of each group by a perfectly arbitrary number given
in col. (5) ; the numbers selected were taken quite at random from

AVERAGES 31

another column of figures in another Blue-book, and had no coa-
nection whatever with the subject of rents ; this gives : —

27(66-0)+54(58-5)+ . . . +55(51-7)
27 + 54 + . . . + 55^"

The last mean, 57-6, is obtained by choosing as weights any
numbers (and for simplicity we choose the smallest) as in col. (6)
which are very roughly proportional to the arbitrary weights used
in the last instance ; we thus get : —

3(66-0)+6(58-5)+ . . . +6(51-7)
3~T 6 + . . . + 6~'

Now the first of these means is clearlj"^ the most satisfactory, since
it is the result of very properly weighting the mean rent of each
group of towns accordhig to the number of towns the group con-
tains. But the second result shows that if we are ignorant of the
number of the towns in each group we shall not be very far out in
our calculation if we treat them all as of equal importance, and find
the simple arithmetic mean of the mean rents in the nine groups.
We can even go further, for we find, from the third and fourth results,
that by weighting the mean rents in the various groups on quite a
random basis, the mean we get still does not differ very greatly from
the best value first found.

The important princi2:>le of which the above example is an illus-
tration is perfectly general, and may be stated as follows : If the
total number of measurements or observations be not very small,
and if the resulting values of the organ or character measured
(rent in our case) be not very unequal, any reasonable selection of
multipliers or weights (as, for instance, the first two adopted above)
will give means which differ from one another by but little ; and
even an apparently unreasonable selection of multipliers (as, for
instance, the third adopted above), assuming they are not so
wildly chosen as to give any particular group a very unfair weight
in comparison with the others, will not throw the mean out badly.
Further, in place of a set of large multipHers we may substitute
small numbers which are roughly proportional to them (as we have
done in the fourth case above), and the mean will again be very
little affected. [See Appendix, Note 2.]

CHAPTER V

AVERAGES {continued)

Applications of Weighted Mean. In determining the weighted mean
of a set of obs(}rvations it is usual, of course, to weight each observa-
tion according to its importance, though what number should be
chosen as a measure of its importance may sometimes be a matter
of doubt. It is not a very difficult matter to decide when we
wish, for example, to compare birth, marriage, or death rates in
two districts, if we know how the constitution of the population
in the one district differs from that in the other, for the weighting
in each of these cases must be in proportion to the population
concerned, and it is too important to ignore.

Death rate, crude and corrected. Imagine a city in which the
total number of deaths in a certain year is N out of a population
numbering P.

The ordinary or crude death rate for that city wiU then be

N

— X 1000, by defuiition.

Now this number N may be analysed according to the ages of
the people who have died ; let us suj^pose it is made up of

Wj people between limits 0 and less than 5 years of age,

'^2 '» !> )> ^ " ^^ »»

*^3 " " " 1^ >' ■^^ »>

and so on, where

^1 + ^2 + ^3+ ... =N.

Again the number P may be analysed accorcUng to the ages of
the people who compose the total population, giving, saj^

jp^ of the population between limits 0 and less than 5 years of age,

Vz J) )) >> >> 5 ,, 15 ,,

Vz " " " '> 1" " ^^ "

and so on, where

Vi-\-Vz-\-Pz+ ■ • • =P-

AVERAGES 33

Thus we may write for the crude death rate

N
D=--XlOOO
P

_W, + W2 + W3 +

X 1000

=*Lii000+*^1000+-n000+ . . ;

=^/!!i 1000 V^f-^ loooV^f "^ loooV . . .

= (Pl<^l +^2^2+^3^34- • • O/P.

where d^is the death rate between limits 0 and less than 5 years of age,
^2 »> " " ^ '» *". >>

and so on.

Now if we compare this expression with the corresponding one for
another city, say,

it is quite conceivable that the death rates in the various age groups
might be equal —

d^=(i , d ^d , d =d . . .

I l' i! 2' .! 3

and yet D might exceed D' because in the first city there are a
greater proportion of infants or old people, on which classes the
hand of death falls heaviest, that is, because the ^'s or weights
wliich multiply the biggest d's are greater in the first case than in
the second. But so long as the d's in the two cities are equal, age
group by age group, it would be reasonable to regard the cities as
equally healthy, or unhealthy as the case might be, and therefore
to insure a fair comparison it is usual in the Reports of the Registrar-
General to give a corrected death rate in place of the crude death
rate defined above.

This is done by weighting the death rate for each age group, not
in proportion to the actual number of persons in that group in
the city itself, but in proportion to the corresponding number in

C

34 STATISTICS

the country at large. Thus, if we denote the proportion of the
population, Q,

between limits 0 and less than 5 in the country at large by QilQ,

„ 15 „ 25 „ „ „ qJQ,

and so on, we get as the corrected death rate

a form wliich has the effect of making the results agree in two
cities which have equal d's throughout.

A similar method of correction is clearly applicable in consider-
ing the incidence of the death rate when we are concerned not
with a difference of district but with a difference of sex, occupation,
religious profession, wage-earning capacity, or any other well-
defined character. Further, it may be used also in comparing birth
rates, marriage rates, heights, weights, chest measurements, or any
similar attributes, when it is necessary to refer the observations
or measurements to a standard population in order to avoid
complications due to age variation.

There is another method of correction, equally general in applica-
tion, which is useful when the death rates in the various age groups
are not laiown. In this case D, the crude death rate for the whole
population of the district is known, alsopJ'P, pJP, P3/P, ■ . • the
proportions of the population between the various age limits, but
d^, do, d^ . . . are supposed unknown.

Now if the population in the country as a whole were the same in
corresponding age groups as it is in the district under consideration,
we should get as the death rate for the whole country

where 8^, S,, S3 • • • are the death rates in the various age groups in
the country at large, and these would in practice as a rule be known.
The actual death rate for the whole country is, however,

{qA+Q2K+Qzh+ • • • )/Q.

where g'j/Q, q^lQ, 33/Q • • • denote, as before, the real proportions
of the population in the various age groups in the country at large.
We take as the corrected death rate required for the district a
number bearing to the crude death rate the same ratio as

i(lA-\-Q2^2+ ' • O/Q bears to {PiSi+jy-^S^-^ . . .)/P.

AVERAGES

Hence we have

corrected death rate qi^i-hQi^^-^-

35

D

PA+P2^2 +

X

Q*

Index Numbers to compare Household Budgets. Another highly
important illustration of a weighted mean occurs in the search for a
satisfactory measure of the change in the cost of living from year
to year. We have already introduced the subject of variation in
wholesale jirices, and we have seen that Sauerbeck, in forming his
index numbers, treats as one each of the forty-five commodities
he uses to measure this variation : the observations, that is to
say, are not weighted.

But, confining our attention to food alone, supposing we have
five items, such as bacon, bread, tea, sugar, milk, for Avhich the
index numbers of prices at two different dates are : —

Bacon.

Bread.

Tea.

Sugar.

Milk.

First date
Second date

100
117

100
95

100
94

100
102

100

109

Is it really right to treat each of these items as of equal importance
with the rest, or ought we to regard bread and tea, say, as of more
weight than bacon, and count bread perhaps five times and tea
three times wliile counting bacon only once ? It is clear that, in
order to select a reasonable set of multipliers in this case, we should
need to know the standard of living of the class of people under
consideration, and how much in the aggregate they spend upon
bacon and how much upon bread, etc.

A partial answer to these questions can be obtained by maldng
a collection of household budgets as was done, for example, by two
Government Committees which recently reported (1918-19) on the
Cost of Living among the Urban and the Agricultural Working Classes
respectively. If the number of commodities employed is large,
even an arbitrary set of multipliers, as we have indicated, will not
displace the mean any great distance from the value when reason-
able weights are chosen, but unfortunately in collecting such house-
hold budgets we are confined to the comparatively limited variety
of food- stuffs which are in general use.

Different principles may be followed in making the comparison

36

STATISTICS

between one year and another which may be illustrated by a few
figures from the Urban Classes Report (1918) : —

Table (10). Household Budgets showing Prices of each Com-
modity AND Quantities Purchased at Two Different
Dates by Typical Family.

Commodity.

First year (1914).

Second year (1918).

Xy

«i

a-.i

n.i

Price (pence

No. of lb.

Price (pence

Xo. of lb.

per lb).

bought.

per lb.)

bought.

Sugar

2-2

5-9

7-07

2-83

Tea .

21-3

0-68

33-3

0-57

Potatoes

0-7

15-6

1-25

200

Let a*i be the price, in pence per unit, of any one commorlity
at the first date, and let n^ be the number of units of this commodity
bought per week by a typical family {n may be estimated in different
ways, e.g. (1) by dividing the total number of units bought by
all families by the total number of those famihes, or (2) by ranging
the different amounts bought by different families in order of
magnitude and picking out the median amount, or (3) by choosing
the mode, i.e. the amount most commonly purchased). Also let x^
be the price, in pence per unit, of the same commodity at the second
date, and let Wg be the number of units of the commodity then
bought per week by the typical family estimated in the same way
as before.

The actual expenditure, measured in pence, at the two dates
will then be

E{x{n,^) and ^{x^n^)

respectively, where Z{x^n^) simply denotes the sum of expressions
like [x^n^ for all the commodities recorded and ^'(a'gWg) denotes the
sum of expressions like {x.^n.^ for all the commodities recorded,
H, the old English S, being a well-known conventional abbreviation
for ' Sum of expressions like.' Thus, with the numbers in Table (10),
we should have

2'(a:iWi)=(2-2)(5-9)+(21-3)(0-68)+(0-7)(15-6)+ . . .
i:(a:2W2)=(7-07)(2-83)+(33-3)(0-57)+(l-25)(20-0)+ . . .

AVERAGES 37

Taking 100 as the index number to represent expenditure at the
first date, the index number measuring expenditure at the second
date may be formed in any of the following different ways,* which
as a rule, of coiu-se, lead to different results : —

(1) l002:ix,n,)lUix,n,) ;

(2) I00i:{x^n,)/i:{x,7i,) or lOOUix-.n^yZix^n^) ;

(3) l00U{Xjn2)l2:{Xini) or 100i;(.r2W2)/Z'(a:2Wi).

The first of these expressions compares the actual expenditure at
the second date to that at the first date.

The next two expressions take into account directly only the
change in prices ; they compare, not actual expenditures but, the
expenditures at the two dates as they would be if the amounts
purchased at the two dates were the same : the first supposing
these amounts to equal those actually bought at the first date,
and the second supposing them to equal those actually bought
at the second date.

The last two expressions, on the other hand, take into account
directly only the change in amounts pui'chased ; they compare
the expenditures at the two dates as they would be if the prices
ruling at the two dates were the same : the first supposing these
prices to equal those actually charged at the first date, and the
second supposing them to equal those actually charged at the
second date.

The particular method of weighting adopted must naturally
de|3end upon the circumstances of the period under discussion
and the nature of the inquiry one is making ; it is a nice question
to decide how far emphasis should be laid upon the old standard
of life (measured by food, lighting, rent, recreation, etc.) with the
expense required to maintain it, and upon the new standard of life
and the cost necessary to reach it.

It may be useful here to summarize a few of the questions of
interest which present themselves in connection with the formation
of index numbers of prices designed to measure changes in the
value of money in general without reference to any particular class
of the community : —

1. What years should be selected in fixing our standard prices ?

2. What commodities should be chosen as a basis for our
average ?

[* See also The Measurement of Changes in the Cost of Living, by A. L. Bowie}', Sc.D.,
in the Journal of the Royal Statistical Socictij, May 1919, for a more complete dis-
cussion of the subject.]

38 STATISTICS

3. What weight should be given to each commodity in relation
to the rest ?

4. How should the prices of the several commodities be deter-
mined, bearing in mind that ' price ' itself frequently varies from
place to place ?

5. Finally, how should these prices be combined to give the
average required ? Should we use the simple arithmetic mean, the
geometric mean [see Appendix, Note 3], the median, or some other
measure ?

While we are not prepared to attempt to answer these questions
fully, seemg that authorities are not altogether agreed as to what
the answers should be, one or two points may be worth noting.
Generally speaking we may say that : —

1. The 3'ears selected m fixing our standard prices should be
years in which economic conditions were normal rather than
abnormal.

2. The commodities chosen should be articles of general con-
sumption, and as wide a field as possible should be covered in their
choice.

3. Many consider that little is gained by weighting, but, if
weights are introduced, the greater the importance of any com-
modity in relation to the rest, judged for example by the relative
quantity consumed, the greater should be the weight assigned
to it.

4. The practical difficulty of assessing retail prices when they
are uncontrolled compels us in general to fall back upon whole-
sale quotations, on wliich some light may be thro^ra by keeping
under observation the important markets for the sale of each
commodity.

5. The average commonly used is the simple arithmetic or the
weighted mean, though arguments can be adduced in favom- of
other averages such as the median.

Leaving index numbers now on one side and returnuig to the
general subject of averages, we may remark that the question
which average is correct in any given case, the mean (weighted or
otherwise), the median, or the mode, does not arise : no one average
is more correct than another, because they are all entirely con-
ventional and represent different ideas ; they correspond in fact
to so many different ways of summing up a set of observations or
measurements in a single numerical statement, and the real question

AVERAGES 39

to determine is which statement, which kind of average, brings the
set of observations before us to the best focus.

For this purpose one average will clearly be best in one case and
another in another, but it may be stated without hesitation that
the ai'ithmetic mean is certainly the most useful of the three and
it is the most frequently used. Other averages have been sug-
gested, such as the geometriG and the harmonic means [see Appendix,
Note 3] familiar to students of Algebra, as suitable in special
classes of problems.*

In a reasoyiably symmetrical distribution of observations, one in
which the variables of medium size arc the most frequent and the
frequency diminishes about equally on either side towards the
largest and the least of the variables, the values of the mean, the
median, and the mode will be found to lie all very close together ;
and a useful jsractical rule to remember is that the median comes
in general between the mean and the mode, the difference between the
mean and the mode being about three times the difference between the
mean and the median. This rule, for lack of a better, might be used
to determine the mode in suitable cases, or it might be used to test
the value found in some other way.

The general term ' average ' is frequently used when the par-
ticular denomination ' arithmetic mean ' is implied, but the context
will usually prevent misunderstanding.

In order to get a clear impression of the outstanding features
presented by the three chief averages discussed, let us go over them
once more in the case of marks awarded to a number of students
in a class. All three may be regarded as in a sense measures of
the standard reached by the class as a whole in the examination,
but the measures are made in different waj's : —

1. The Arithmetic Mean is found by merely dividing the aggregate
marks of the class by the number of the students, and it gives the
marks earned by each student if we conceive them all to be of
equal merit.

2. The Median is found by ranging the students in order of merit
from top to bottom, and picking out the marks awarded to the one
who comes half-way down the list.

3. The Mode is the most fashionable number of marks, i.e. the
marks obtained by the greatest number of candidates.

The advantages and disadvantages of the three types may be
set out broadly .as follows, although the boundary lines must not
be too strictly drawn : —

* See Note on p. 41.

40

STATISTICS

Mean.

Median.

1
Mode. j

1

Easy to calculate when

Easy to pick out when

Not easy to determine

the values of the vari-

the individuals can

with precision, when

able can be summed

be ranged in order

the observations faU

and their number is

according to the

into groups of differ-

known.

value or degree of

ent ranges, without

the variable ob-

fitting a frequency

served.

curve to the distribu-
tion as a whole.

Well designed for alge-

Unsuited for algebrai-

Unsuited for algebrai-

braical manipulation,

cal work.

cal work.

as, for example, when

we wish to combine

different sets of obser-

vations [see Appendix,

Note 4, for two illus-

trations].

Affected sometimes too

Determined merely by

Unaffected by abnor-

much by abnormal in-

its position in the

mal indiAaduals, and

dividuals among the

distribution, and its

owes its importance

observations.

actual value is thus

to the fact that it is

quite unaffected by

located in the region

abnormal individuals.

where the frequency
is most dense.

The reader should test his grasp of the principles so far intro-
duced by applying them himself to a concrete case. For exami^le,
he might use the data in Table (11), with regard to wages earned
by certain women, taken from Tawney's Minimum Wages in the
Tailoring Trade, and based upon the 1906 Wages Census. Let him
begin by roughlj^ estimating the mean, the median, and the mode
from an inspection of the distribution. He might then proceed
to calculate the mean wage : —

(1) taldng the actual frequencies given in the table ;

(2) taking simple sub-multiples of these frequencies, roughly one-

hundredth part of each : 2, 4, 6, 7, 9, 11, etc. ;

(3) assuming unit frequency in place of that given in the table for

each wage grouj).

Finally, he might determine the median and the mode in the
manner explained in the text, deducing the latter from the relation
(mean— mode)— 3(mean— median).

AVERAGES

The results obtained should be

(1) 13-08S. ; (2) 13-lOs. ; (3) 15-59s.
Median=12o3s. : Mocle=ll-43s.

41

Table (11). Distribution of Wages of certain
Women Tailors.

(1)

(2)

(3)

(4)

No. of "Women

j
No. of Women

Wages between limits

earning wages
as shown in

Wages between limits

eaniiiig wages
as shown in

Column (1).

Column (3).

5s. and less than 6s.

180

16s. and less than 17s.

642

6s. ,

7s.

384

17s. „

18s.

453

7s. ,

8s.

553

18s. „

19s.

401

8s. ,

9s.

690

19s. „

, 20s.

272

9s. ,

10s.

900

20s. „

21s.

251

10s. ,

lis.

1145

21s. „

, 22s.

138

lis. ,

12s.

1201

22s. „

23s.

124

12s. ,

13s.

1138

23s. „

24s.

64

13s. ,

14s.

930

24s. „

25s.

54

14s. ,

15s.

885

25s. „

, 30s.

122

15s. ,

'

16s.

790

..

* [One important example of the use of the geometric mean is in the con-
struction of the Board of Trade Index Number of Wholesale Prices— see article
by A. W. Flux, C.B., M.A., in the Journal of the Royal Statistical Society,
March 1921.]

CHAPTER VI

DISPERSION OR VARIABILITY

Let us suppose that two men set out separately on walking tours
and that they walk as follows : —

The total distance covered in six days, namely 150 miles, and
therefore also the mean rate of walking, 25 miles a day, are thus
exactly the same in both cases, but the disjJersio^i of the values of
the variable (the variable being in this instance the number of
miles walked per day) round about their mean value, the variability,
is different in the two cases. The greatest deviation from the
average in the fii'st case is five and in the second case it is ten miles.

Thus, besides knowing the average of a set of values of a variable
it is important to measure the dispersion of the distribution. Are
the observations crowded in a dense mass around the average,
or do they tail off above and below it, and to what extent ?
In other words, what is the variability from the average of the
distribution ?

Mean Deviation. Now we are not concerned here with the signs
of the separate deviations, with the question, that is, whether any
particular value of the variable lies above or below the average :

DISPERSION OR VARIABILITY

43

it is only of their amount we wish to take cognizance, and perhaps
the most obvious Avay to measure the total variability and at the
same time to ignore the signs of the separate deviations from the
average is to add up these deviations, treating them all as signless,
and to divide the result by their total number. This gives what
is known as the mean deviation of the system of observations — it
is the ordinary arithmetic mean of the separate deviations, treated
as if they are all in the same direction, and, in measuring them, we
may use either the mean or the median as the average, but it
would seem preferable to take the latter because the mean deviation
is least when the median is chosen as the origin, or zero point, from
which the differences are measured. The proof of this fact will
be found in Note 6 in the Appendix, but we may readily test it in
a given case.

Let us adapt the ' walking ' illustration used above, slightly
extending the figures and making them unsymmetrical, i.e. of
unequal variability on either side of the average, so as to prevent
the median coinciding with the mean. We then have an amended
table setting out the number of miles walked b}^ a certain man on
successive days during, sa^^, a fortnight's tour, as follows : —

Table (12). Number of Miles walked on Successive Days.

(1) (2) (3) (4) (5) (6) (7) (8)

No. of

(lays.

Miles
walked.

X

Deviation
from 25.

^1 r.
Deviation; j)^^-^j^„

|-- from 24.

Xi

Deviation
from 2G.

fx

[No. in

Col. (1 )] X

[No. in

Col. (3)].

M

[No. in

Col. (l)]x

[No. in

Col. (4)].

1

2
3
3
2
2
1

14

10
15
20
25
30
35
40

15

10
5

5
10
15

14-64
9-64
4-64
0-36
5-36
10-36
15-36

14

9
4
1
6
11
16

16
11
6
1
4
9
14

15
20
15

10
20
15

14-64
19-28
13-92
1-08
10-72
20-72
15-36

••

••

95

95-72

The first two columns show that 10 miles was the distance walked
on the first day, 15 miles on each of the next two days, 20 miles
on each of the next three days, and so on until the last day, when
40 miles was the distance walked.

44 STATISTICS

The median in this case, being the number of miles walked on
the middle day when the daj's are ranged in order of mileage from
the least to the greatest, is 25, for this is the distance covered on
both the seventh and the eighth days which come half-way along
the series.

Col. (3) shows the deviations from the median, 25, of the distances
covered each day as recorded in col. (2), and col. (7) enables us to
sum these deviations when each is multiplied by the number of
days to which it corresponds, since these numbers, given in col. (1),
show how many times each deviation is repeated. Hence the mean
deviation, regardless of sign, measured from the median

-[(Ixl5)+(2xl0)+(3x5)+(2x5)+(2xl0)+(lxl5)]/14
= (15+20+15+10+20+15)/14
=95/14
=6-79 miles.

We may compare this with the corresponding deviations measured
from (1) the arithmetic mean, (2) the number 24, and (3) the
number 26 as origin respectively.

1. The arithmetic mean of the distribution is obtained at once
by multiplying the corresponding numbers in cols. (1) and (2),
adding the results, and dividing the total by 14, thus

. .^, . l(10)+2(15)+3(20)+3(25)+2(30)+2(35)+l(40)
Arithmetic mean= - —

1+2+3+3+2+2+1

10+30+60+75+60+70+40

14
=345/14

=24-64 miles,

and the deviations from 24-64 are shown in col. (4) ; the mean
deviation from 24-64, obtained by combining cols. (1) and (4) and
adding as shown in col. (8)

= [i(14-64)+2(9-64)+ . . . ]/14

=95-72/14

= 6-84 miles.

2. Similarly, the mean deviation from 24, making use of col. (5),

= [l(14)+2(9)+ . . . ]/14
= 6-93 miles.

DISPERSION OR VARIABILITY

45

3. And the mean deviation from" 26, making use of col. (6),

-[1(16)+2(11)+ . . . ]/14
=7-07 miles.

The original determination gives a value which is less than any
of these three results, as was anticipated.

The mean deviation from the median is, however, difficult to
calculate with exactness when the observations are recorded in
groups between different limits : for this and other reasons we
shall not spend much time upon it, and we shall as a rule choose
the mean as origin of reference rather than the median. It
may be as well to explain the source of the difficulty by a small
hypothetical illustration.

Let us suppose that in making measurements of some organ or
character in 13 individuals we get a result l}dng between 4 and 6
units on six occasions, between 6 and 8 units on four occasions, and
between 8 and 10 units on three occasions. Here, assuming that all
the individuals in any group have the mid-value measurement for
that group, i.e. treating the distribution as one of 6 individuals
with a variable measuring 5 units, 4 individuals with a variable
measuring 7 units, and 3 individuals with a variable measuring
9 units, we get ^ as the mean deviation with 7 as origin and ^
for the mean deviation with 6-5 as origin, as the following table
shows : —

Now the result obtained is in agreement with the minimum
mean deviation theory, granted that 7 is the median measurement,
as it might certainly be. But it is not so of necessity, and in that
case the assumption italicized might lead, in the above calculation,
to appreciable inaccuracy unless the number of observations is
large and the class-interval is small. For example, the actual

46

STATISTICS

distribution might, without contradictmg the previous data, con-
ceivably run : —

Measurement.

/'

Frequency.

Deviation
from 7.

y'

Deviation
from 6-5.

fx'

fy'

5

6-5

7-5
9

6

2
2
3

2

0-5
0-5
2

1-5

1
2-5

12

1
1
6

9

2'

7-5

18-5

13

••

••

20

But in this case the median, the measurement for the seventh indi-
vidual from either end of the series, is 6-5, and according to the
first calculation the mean deviation referred to 6-5 as origin appears
to be greater than that referred to 7 as origin. If, however, we
recalculate, using the more detailed table, we find that the mean
deviation referred to 6-5 as origin (^^) is really less than the mean
deviation with reference to 7 as origin, as it should be, for the
latter now turns out to be %.

Standard Deviation. An alternative method of avoiding the
signs of the deviations from the average in order to estimate the
amount of variability of the distribution is to square each separate
deviation, sum the squares, divide by their number, and take the
square root of the result. This gives the root-mean-sqimre deviation,
and it is least when the arithmetic mean of the variables is chosen
as origin from which to measure the deviations, when it is known
as the standard deviation. For proof of this minimum principle
see Appendix, Note 5, but it is worth while testing it also with the
data given in Table (12).

The numbers in cols. (3) to (6) in Table (13) are obtained simply
by squaring the corresponding numbers in the same cols. (3) to (6)
in Table (12). Col. (7) is formed in order to enable us to calculate
the mean-square deviation referred to 25 as origin ; the numbers
in col. (3) show the squares of the deviations for each individual
observation, and the numbers in col. (1), by which they are multi-
plied, show how frequently the same values are repeated. Hence
we get the mean-square deviation with reference to 25

= [l(225)+2(100)+3(25)+2(25)+2(100)+l(225)]/14

=975/14

= 69-64.

DISPERSION OR VARIABILITY

47

Thus the root-mean-square deviation referred to 25

= V(69-64)
= 8-345.

Similarly, by means of col. (8), formed on exactly the same
principle, we find that the root-mean-square deviation referred to
24-64 as origin

= V'[(214-33+ 185-86+ . . . )/14]

= V(973-22/14)

=8-338.
But 24-64 is the mean of the distribution, hence 8-338 is the standard
deviation.

With the help of cols. (5) and (6) the student may himself calcu-
late the root-mean-square deviation wdth regard to 24 and 26
respectively as origin ; the results should be 8-36 and 8-45. Of
the four values thus obtained for the root-mean-square deviation,
the least is that referred to the mean as origin, the standard devia-
tion, now proposed as a measure of variability or dispersion suitable
for most general purposes.

This measure possesses several decided advantages over the
mean deviation ; among others it lends itseK more easUy to certain
algebraical processes (see, for example, p. 158), a fact of importance
when we wish, for instance, to discuss two sets of observations in
combination, and it is in general less affected by ' fluctuations of
sampling ' — errors which arise owdng to the fact that we cannot as
a rule survey the whole field of operations, but have to be content
"with a sample.

Table (13). Number of IMiles walked on Successive Days.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

/

No.

of

da3's.

Miles
walked.

Square of
Deviation
from 25.

Square of
Deviation
from 24 -64

Square of
Deviation
from 24.

Xi

Square of
Deviation
from 20.

fx-2

[No. in Col (1)]

X

[No. in Col. (.3) j

[No. in Col. (1)]

X

[No. in Col. (4)]

1

10

225

214-33

196

256

225

214-33

2

15

100

92-93

81

121

200

185-86

3

20

25

21-53

16

36

75

64-59

3

25

. .

0-13

1

1

0-39

2

30

25

28-73

36

16

50

57-46

2

35

100

107-33

121

81

200

214-66

1

40

225

235-93

256

196

225

235-93

14

••

975

973-22

48 STATISTICS

Quartile Deviation or Semi-interquartile Range. There is a third
measure of dispersion, based upon the determination of the quartiles,
and to introduce them we may refer again to Table (7) in order to
show how the idea of the median may be extended.

We define the individual occupying a position one -quarter the
way along any series of observations, arranged in ascending order
of magnitude of some organ or character common to all the indi-
viduals of the series, as the lower quartile ; and we define the indi-
vidual occupying a position three-quarters the way along the series
as the upper quartile.

When the distribution of observations is divided up into groups
lying between different limits of the variable under consideration
the quartiles may, like the median, be calculated by interpolation.
Thus, in the examination example, the total number of candidates
is 514 and i(514)=128-5.

But the 91st candidate from the bottom gets approximately 20
marks, and the 149th candidate from the bottom gets approxi-
mately 25 marks. Hence the imaginary candidate, No. 128-5,
should get a number of marks lying somewhere between 20 and
25. But if, in this neighbourhood, a difference of

(149-91) candidates corresponds to a difference of 5 marks,

37'5

(128-5-91) ,, should correspond ,, 5x marks.

Thus, the marks assigned to the lower quartile candidate are
approximately

58
=20+3-23.
Hence the lower quartile=2S-2d.

Again |(514)= 385-5.

But the 318th candidate from the bottom gets approximately 35
marks, and the 397th candidate from the bottom gets approxi-
mately 40 marks. Therefore, the imaginary candidate, No. 385-5,
should get approximately a number of marks

79
= 39-27.
Hence the upper quartile=39-2'7 .

DlSl'ERSION OR VAlilABlLITiT 4d

It is clear that the quartiles together with the median divide the
whole series of observations into approximately four equal groups, so
that the quartile marks
give a rough idea of the 23'23 31 '52 39'27

distribution on either O j;j^^] — q;:

side of the average. For

this reason half the difference between the quartiles provides a
convenient measure of the dispersion, and it is called the quartile
deviation or semi-interquartile range ; thus, if Q be the lower and
Q' the upper quartile, we have

the quartile deviation=^\{Q' —Q).

In the above example, this measure

= i(39-27-23-23)
= 1(16-04)
= 8-02.

If a more minute analysis of the distribution of variables is
desired, we may range them in order of magnitude as before, and
divide up the series into ten equal parts, recording every tenth along
the line ; these tenths are called deciles.

Thus, the deciles in the examination example correspond to the
marks assigned to imaginary candidates numbered as follows : —

51-4, 102-8, 154-2, 205-6, 2570, 308-4, 359-8, 411-2, 462-6,
and they can be calculated by the interpolation method used in
fuiding the median and quartiles.

This way of representing the chief features of a distribution, by
quartiles, etc., was much used by Galton in his researches and
writings.

The student mo^y be perplexed as to which should be used of so
many different measures of dispersion or variability, but there
need be no real confusion. If a rough estimate only is wanted the
quartile deviation is a convenient measure, assuming that the
variables observed or measured can be ranged in order of magnitude
so as to admit of the quartiles being readily picked out. Also the
measure thus obtained is not unsatisfactory when the distribution
of values of the variable is fairly symmetrical and uniform in its
gradation from greatest frequency to least. If, however, it is
conspicuously skew (unsymmetrical) and there are erratic differ-
ences in frequency between successive values of the variable, it
is better to choose a measure which gives the magnitude and
the position of each recorded observation its due weight in the
deviation sum.

D

50 STATISTICS

Then again the choice as between the standard deviation and the
mean deviation may be sometimes determined by the particular
kind of average which suits the problem best. But as the arith-
metic mean is the most important and the most commonly used
average, so the standard deviation is certainly the most important
measure of dispersion.

It will be shown later that the following relations are approxi-
mately true when the distribution of variables is not very far from
being symmetrical : —

(1) Quartile deviation=^ ^{Standard deviation).

(2) Mean deviation =i{Standard deviation).

In (2) the mean deviation should be measured from the mean.

Also (3) a range of two or three times the standard deviation
on both sides of the mean Avill be found to include the majority
of the observations in the distribution.

Coefficient of Variation. Before \Ye pass on to illustrate the
subject of averages and variability by means of a few examples
it is necessary to introduce one more constant known as the co-
efficient of variation. It is a measure of variability but it differs
from the chief measures alread}^ discussed in that they are absolute
measures, whereas the coefficient of variation, wTitten C. of V. for
short, is a ratio or relative measure. The need for it arises when
we reflect that in order to gauge fairly the amount of variability we
ought to have in mind also the size of the mean from Avhich the
variation is measured ; just as a difference of 1 foot between the
heights of two men is a conspicuous difference when the normal
height is between 5 and 6 feet, whereas the same difference of 1 foot
between two measured miles would be trifling because the standard
mile contains over 5000 feet.

The coefficient of variation has been defined by Karl Pearson
{Phil. Trans., vol. 187a p. 277), who first suggested its use, as ' the
percentage variation in the mean, the standard deviation (S.D.)
being treated as the total variation in the mean,' so that

C. of V. = 100 S.D./Mean.

He pointed out that it would be idle, in dealing with the variation
of men and women (or indeed very often of the two sexes of any
animal), to compare the absolute variation of the larger male organ
directly with that of the smaller female organ, because several of
these organs, as well as the height, the weight, brain capacity, etc.,

DISPERSION OR VARIABILITY

51

are greater in man than in woman in the approximate proportion
of 13 : 12.

As an example of the use of the C. of V., figures may be quoted
from a paper by R. Pearl and F. J. Dunbar {Biomelrika, vol. ii.
pp. 321 et seq.), On Variation and Correlation in Arcella. Measure-
ments in mikrons were made of the outer and inner diameters of
504 sjDecimens of a shelled rhizopod belonging to the group Impcr-
forata, family ArcelUna, with the following results, to two decimal
places : —

Mean.

S.D.

C. of V.

Outer diameter .
Inner „

55-79
15-91

5-73
217

10-27 per cent.
13-66 „

Thus, judging by the S.D. column, giving the absolute size of
deviation, the outer diameter would appear to be more variable
than the inner, but the C. of V. column shows that, if we take the
sizes of the two diameters into account, the inner is reaUy the
more variable of the t^^■o. To turn aside the edge of possible criti-
cism it should be added that the authors also give the errors to
which the above measures are subject, as unless these are known
we cannot tell whether the differences observed in variation are
significant or not of a real difference in fact, but that question
must be left until the theory of errors due to sampling has been
developed in a later chapter.

The C. of V. varies considerably for different characters. W. R.
Macdonell states that ' 3 to 5-5 are rei^resentative values for varia-
biUty in man, Avhile in plants it may run to 40,' and Pearson and others
have shown that for stature in man it varies from about 3 to 4
and for the length of long bones from 4 to 6.

CHAPTER VII

FREQUENCY DISTRIBUTION : EXAMPLES TO ILLUSTRATE
CALCULATING AND PLOTTING : SKEWNESS

Calculation of Mean and Standard Deviation. Example (1). — We
return now to the examination example in order to show how the
labour of calculation in finding the arithmetic mean and standard
deviation of a frequency distribution may be somewhat lessened.

The various steps in the process appear in Table (14). In the
first column the marks at the middle of each class-interval have
been written down, and we make the assumption that all the candi-
dates in any one class have the same number of marks, namely, the
marks at the middle of the class-interval. In any case where the
number of observations is large, and where the class-intervals are
reasonably small, the errors resulting from such an assumption will
be insignificant, because the individuals in each class are Just as
likely to have values above as below the value at the middle of the
class-interval, and they will therefore compensate for one another.

We now seek to alter the scale of marking so as to produce a
simpler set of marks than the original, which will make the work
of finding the mean also simpler, but we must not forget at the
end to change back again to the original scale. We choose a number
from col. (1), somewhere near the required mean, to act as a land
of origin from which to measure the other numbers in the column.
This choice is only a rough guess, and it is really immaterial which
number is selected as origin, except that the nearer it is to the
mean the lighter will be the calculation to follow ; the number 33
has been selected in this instance.

In col. (2) are written down the deviations of the marks in each
class from 33, so that now some candidates appear as if they were
6, 10, 15 . . . marks to the bad, and others as if they were 5, 10,
15 ... to the good. So long as we remember to add 33 at the
end we can content ourselves therefore by finding the mean of the
marks as given in col. (2). But these again can be further simphfied
by dividing each candidate's marks by 5, and we then only need

52

FREQUENCY DISTRIBUTION 53

to find the mean of tlie marks as shown in col. (3), so long as we
remember to multijDly by 5 at the first step back to the old scale
of marking. The addition of col. (5) makes it easy to calculate
this mean, for it gives the result of multiplying each value of the
variable (the number of marks in each class) by its appropriate
weight (the number of candidates who obtained that number of
marks).

Table (14). Marks obtained by 514 Candidates in a certain
Examination — (Analysis of Method for Calculating
Mean and Standard Deviation).

Thus, on this new scale, the mean marks obtained are

5(_G) + 9(-5)+2S(-4)+ . . . +87(0)+ • • • +6(+5) + 3(+6)

514

-532+422
5l4

-110

514
:-0-214.

54 STATISTICS

This, then, is the mean of the marks obtained by the candidates on
the scale indicated in col. (3). If the marks are on the scale given
,in col. (2), the mean is 5(— 0-214), i.e. —1-070. To bring them back
to the original scale as in col. (1) we must add 33 to this result, so
that the required arithmetic mean

=-33+5(-0-214)
= 33-1-070
= 31-93.

To find the Standard Deviation, or the root-mean-square deviation
from the arithmetic mean, it is convenient as before to work with
the simplified scale, to measure the deviations from the arbitrary
origin (33) associated with that scale, and to make the necessary
corrections at the end of the work.

Col. (5) in Table (14) gives the deviation multiplied by tlie
frequency in each class, the frequency denoting the number of
times the particular deviation occurs. Hence, if these numbers be
multiplied again by the numbers in col. (3), we shall have each
separate deviation squared and multiplied by its frequency. The
results are shown in col. (6), and they must be added, and their
sum divided by the sum of the frequencies (514), to give the mean-
square deviation, which we may represent by s^.

Thus 52^2814/514

= 5-475,

and this is the mean-square deviation referred to 33 as origin.
We require the corresponding ex23rcssion referred to the mean,
31-93, as origin. If we denote this by s,,^^ there is a simple relation
connecting the two, namely,

where x is the deviation of the mean itself from 33 [see Appendix,
Note 5] ; of course 5„j, s, and x are all to be measured on the same
scale, the simplified scale adopted with 5 marks as unit.

Now we have already shown that the deviation of the mean from
33 — —0-214, and this is therefore the value of x.

Hence ^^) s,„2_5-475- (-0-214)2

^('Y' .r =5-475-0-046
,,S' y =5-429

FREQUENCY DISTRIBUTION 55

And, returning to the old scale, the standard deviation, usually
denoted by a

=5(2-33)

= 11-65.

We notice that 3ct= 34-95, and this ranj^e on either side of the
mean amply takes in all the observations.

The mean deviation is readily found from Table (14) by adding up
the numbers in col. (5) regardless of sign and dividing by the sum
of frequencies, 514.

Thus, on the new scale, the mean deviation

0 54

5 14

= 1-856,

which, on the old scale, becomes 5(1-856) or 9-28. This, however,
is the mean deviation measured from 33 as origin, and a correction
has to be applied to get the mean deviation measured from the
median or from the mean.

To get the mean deviation from the mean we note that the
difference between the mean, 31-93, and 33 is 1-07. Hence it
should be clear from Table (14) that, by measuring from 33 instead
of from 31-93, Ave have made the deviations of all the marks from
33 upwards too little by 1-07, and we have made the deviations of
all the marks frojn 28 downwards too much by 1-07. Hence, to
get the deviation required we must add to 9-28 an amount

= 5T4[l-07(87+79+ . . . +3)- 1-07(82+58+ . . . +5)]

1-07
=iJi: (283-231)
514

• = x52

514

=0-108.

Therefore, the mean deviation measured from the mean=9-39.
This may be compared with ;, (standard deviation) =9-32.

Also the quartile deviation _f or this distribution has been shown
to be=8-02, and it may be compared with |(standard deviation)

=7-77.

Plotting of a Frequency Distribution. The data for the two
examples which follow are taken from the Quarterly Return of
Marriages, Births, and Deaths, No. 261, issued by the Registrar-
General.

56

STATISTICS

The first shows the proportion to population of cases of infectious
disease notified in 241 large towns of England and Wales for the
thirteen weeks ended 4th April 1914. This proportion was given
for each town sei^arately in the Return, but, in order to bring out
the distinctive features of the distribution, the several towns have

Table (15). Proportion to Population of Cases of Infectious
Disease notified in 241 Large Towns of England and
Wales during the Thirteen Weeks ended 4th April 1914.

Case Rate
per 1000
persons
living.

Each dot below represents One Town with Notified Rate of Infectious Disease
between limits as given in previous coUinin.

Total No.

of Towns

with given

Rate.

0—
2—
4—

6—
8—
10—
12—
14—
16—
18—
20—
22—
24—
26—

5

39

69

41

29

22

16

7

5

3

4

0

0

1

241

been, in Table (15), represented by dots and put into different classes
according to the proportion of infectious cases notified in each,
with a separate line for each class : e.g. if the proportion for any
town was 5-37 a dot was placed in the line corresjDonding to the
class of to^\'ns for which the rate was ' 4 and less than G.' Every

FREQUENCY DISTRIBUTION

67

fifth dot in each line was ticked off, so as to make them easy to
count up and also to keep the lines, down the paper as well as
across, straight. The frequency, i.e. the number of dots in each
class, Avas then recorded in a column at the extreme right-hand
side of the paper.

70

fc,65

60

to 55

50

§45

'tr..

?40

;35

s:30

W25

=> 20

a^lO

03 5

o

: :; ± --

i

1 1 ' j 1 1

, —

-j |-i —

) — ]

^__L

, —

( ' —

, , —

U-H^-^

A - "\

,

• j '

, 4 1 1 ' 1

• f~ -•

4^ _^_ _i 1

^ _^ ^

L-i — o--* 1 [— p-l 1 1 j j

It ::: :: ::: :: : :::::::

-f- 4 -I -•-

j-j-«--2----j-n |--i 1

«- -»_ -4-L4

»_-, ^p4 — 4--«

n- -1 - -4- -4- -4

-p-4--4--i — 4--i 1

4J^4n-» — *--• l--r-T h-

0- -o-p4- -4- -4 — 1 1 1

4_-4_L4__4_^4-L,

»-'-*- » i « -»--« 1 —

-^' -?-»"-• T

4- -if- -»- _»- -»_ -« ]

— L4_ _«_. _4_ _4- _4_ _4 1

-__4__n--4--«--r-4--XX

-^-f-p-j--4— 4--4-p-4 1

-•--•--I i---»--«--»--»--i -1-9- -•- -•

0 5 10 15 20 25 30 ,x

Rate of Disease per 1000 persons living
Fio. (1).

It will be at once seen that this procedure, without calculating
any averages, etc., ultimate!}' gives to the eye a very good picture
of the distribution, and indeed it is the basis of the graphical method
of studying statistics. In drawing a proper graph we use a specially
ruled sheet of paper which is divided up into a large number of
equal small squares by ' horizontal ' (cross) and ' vertical ' (up-and-

58

STATISTICS

down) lines. This merely enables us to j)lace our dots accurately
in position, as shown in fig. (1), where the numbers 0, 5, 10 . . .
have been marked off along the Hne Ox to correspond to * case

y

70

CD

vC,65

S 60

§55

CD
.03

.o

Ho
O

^45

40

C5.

S35

'5>30

25

2 20

o
§"10

0:

, L

I ^ Modal Line

^/

t ^^

t t^^

Y

^^ 1

..^ ^^ ir

X^ it

. . A

t /'^

_ . . ^S"

^^1

I

\ ±

n

jr

X

1

"t I

t A ^

1.

v.^

\I

± -^ / X

^^t

" " T 3 IL

"it 'T~ % X

3-.-^! +__

20

25

Rate of Disease per 1000 persons living
Fio. (2).

rates ' of these magnitudes : thus rates of ' 4 and less than 6 '
were recorded by 69 successive dots along a vertical line at a dis-
tance 5 (the centre of the class-interval 4-6) from the axis Oy.

FREQUENCY DISTRIBUTION

59

The final configuration in fig. (1), when turned half round, is
exactly the same as that of Table (15). K desired the frequency

y

X ip T

^

"S

•♦r^

S nn

% ^°

,"2 4-

O

"5 KC

g ^^

.O

■«o

^AK

C^45

^*"

'^

Modar tme

= An

.,, ^^

o 40

"■? "*

^ -j- M ' ; dian- Ijrine

Q.

■ ' 7 ^^

P -3 =

,f ■ ^

y '' Vif T •

; — ^/ M ;an i±ine

S

'1 J^

T /

•2 __

; ■ ' 1

-T-- , 1 _r _j_

05

i 1 i"

-c

■"^

<:

~J"

i2

*K

2r> '5

— 1—

o

s

a

^ ^n

QJ 10

u;

5 ■*-

-•- 1

■*" 1

-^ ^ ,1

n -

t_:i,:._: , .TLTti-i::!:--

O 5 10 !5 20 25 30

Rate of Disease per 1000 persons living
Fio. (3).

may be recorded, dot by dot, on a side piece of paper and then
only the topmost dot in each class need be marked on the graph
sheet. In order, however, to enable the eye to measure the height

60 STATISTICS

of each frequency in relation to the rest, it is advisable in that
case to connect up adjacent dots as in fig. (2) or as in fig. (3).

The last method of representation (fig. (3)), to which the name
histogram has been given by Professor Karl Pearson, is particularly
useful and should be carefully studied. It is formed in this case by
erecting a succession of rectangles with the lines 02, 24, 46 . . .
along Ox as their bases, corresponding to the successive classes of
the given distribution, and with heights proportional to the fre-
quencies proper to those classes. It is not necessary to complete
the sides of the rectangles, but, if they were completed, each would
enclose a number of squares proportional to the frequency of towns
■with the rate of disease defined by its base : e.g. the first rectangle
would enclose 10 squares, the second 78, the third 138, and so on,
numbers respectively proportional to 5, 39, 69, and so on. It
follows that the total area enclosed between the histogram and the
axis Ox is proportional to the aggregate frequency of towns observed.

Now we might conceive a step further taken and a smoothed
curve drawn freehand so as to agree as closely as possible with
fig. (2) or fig. (3), but with all the sharp corners smoothed out, and
so nicely adjusted as to make the area enclosed between the curve,
the axis Ox, and lines parallel to Oy defining the limits of any class,
proportional to the frequency of towns in that class. To this
fig. (2) and fig. (3) might be regarded as aj)proximating if only a
sufficient number of observations were recorded, and only in that
case would it be possible to draw it with any accuracy. Such a
curve is called a frequency curve, measuring as it does the frequency
of the observations in diiferent classes.

[Assuming that corresponding to a given frequency distribution a curve
of this kind does really exist — and the assumption turns upon the frequency
being continuous — the reader who is acquainted with the notation of the
Calculus wdll recognise that, if {x, y) represents any point on the curve, ybx
measures the frequency of observations or measurements of an organ or
character lying between the values x and {x-\-bx), when the total frequency
comprises a large number of observations, say 500 to 1000.

Further, it will appear later that the mean, the median, and the mode
have a geometrical interpretation of no small importance associated with the
curve.

The mean x corresponds to the particular ordinate y which passes through
the centroid or centre of gravity of the area between the frequency curve
and axis Ox, because

the mean= J ^ 'S,{x . ybx) j J^ 2(7/5a:),

5.1-^0 S,r->-0

where the summation extends throughout the distribution,

= jxydx/jydx
where the integral extends throughout the curve.

FREQUENCY DISTRIBUTION

61

The median x corresponds to the ordinate y which bisects this same area ;
e.g. in fig. (3), the number of small squares on either side of the median in the
space bounded by the histogram and the axis represents half the total number
of observations, two small squares corresponding to each observation.

The mode x corresponds to the maximum ordinate of the curve, measuring
the greatest frequency in the whole distribution.]

Skewness. There is one feature of a frequency distribution which
catches the eye sooner ahnost than any other, and that is its sym-
metry or lack of symmetry. It is important therefore that we
should have some means of measuring it.

In a sjanmetrical distribution the mean, mode, and median
coincide, and we have, as it were, a perfect balance between the
frequency of observations on either side of the mode or ordinate of
maximum frequency. In a skew distribution the centre of gravity
is displaced and the balance thrown to one side : the amount of this
displacement measures the ske\^^less. But there is another factor
to be taken into account, for when the variability of the distribu-
tion is great the balance is more sensitive than when it is small,
and the difference between mean and mode is consequently more
pronounced though it may not be significant of any greater skew-
ness. This will be clear in the light of the analogy of the swing
of a pendulum. If OPP' denote the pendulum in the accompanying
figure, OAA' its mean position, and OBB' an extreme position, the
displacement in the position OPP' from the mean, if measured
along the scale AB, is AP,
and, if measured along the
scale A'B', is AT'. But,
since the amount of swing
in either case is the same,
it would be more appropri-
ate to write the linear dis-
placement as a fraction of
the full swing so as to make
these two measures also the
same, thus

AP/AB=A'P'/A'B'.

So, in the case of a fre-
quency distribution, Profes-
sor Karl Pearson has suggested as a suitable measure for skewness,
not the difference between mean and mode, but the ratio of this
difference to the variability. Thus

sJcewness= {mean— mode) [ S .D.

62

STATISTICS

or, approximately,

=3(mean— median)/S.D. (see p. 39),

a form which is sometimes useful.

According to this convention the skewness is regarded as positive

Skewness

Skewness

Mode Mean

Mean Mode

-^

X increasing

X increasing

when the mean is greater than the mode, and as negative when
the mode is greater than the mean.

Illustrations of frequency curves, Avith the position of mode and
mean marked, will be found in Chapter xvii.

We proceed to the detailed calculations necessary in the infectious
diseases example.

Table (16). Proportion to Population of Cases of Infectious
Disease notified in 241 Large Towns of England and
Wales during the Thirteen Weeks ended 4th April 1914.

(1)

(2)

(3)

(4)

(5)

Case Rate per
1000 persons living.

Deviation
from 7.

Frequency of
Towns with
given Rate.

Product of

Nos. in

Cols. (2) & (3).

Product of

Nos. in

Cols. (2) & (4).

0 and less than 2

{X)

- 3

(/)
5

(A-)

-15

(A-'-)

45

2 „ „ 4

_ 2

39

-78

156

6^ » » 6

- 1

69

-69

69

6 „ „ 8

. .

41

8 „ „ 10

+ 1

29

+ 29

29

10 „ „ 12

+ 2

22

+ 44

88

12 „ „ 14

+ 3

16

+ 48

144

14 „ „ 16

+ 4

7

+ 28

112

16 „ „ 18

+ 5

5

+ 25

125

18 „ „ 20

+ 6

3

+ 18

108

20 „ „ 22

+ 7

4

+ 28

196

26 „ „ 28

+ 10

1

+ 10

100

••

241

+ 68

1172

FREQUENCY DISTRIBUTION 63

Example (2). — The various averages and measures of variability
of the distribution can be calculated just as in the case of the last
example, and the data required to determine the mean and the
standard deviation are set out in Table (16). We can afford now
to miss out some of the more obvious steps in exjDlanation.

On the scale of col. (2), where a difference of 2 in the case rate,
per 1000 persons living, is the unit and where a case rate of 7 is
taken as origin, the mean, by the result of col. (4)

n s_

— 2 4 1

=0-282.
Hence, on the original scale, the mean

= 7+2(0-282)
=7-564.

Again, the mean-square deviation, on the scale of col. (2), measured
from 7 as origin is

6 241

= 4-863 ;

and X, the deviation of the mean from 7 as origin, on the scale of
col. (2)=0-282. Thus the mean-square deviation measured from
the mean,

=4-863- (0-282)2
=4-783.

Therefore, the standard deviation a, on the original scale

= 2\/'4^783
=4-374.

Since 3cr= 13-122, the range ' (mean— 3a) to (mean+3CT) ' includes
all but one or two observations.

To determine the median, Ave conceive the towns ranged in order
according to the proportion of infectious cases notified in each,
from the least to the greatest, and the town -^dth the median rate
is the 121st from either end.

But the 113th towTi has a notified case rate of approximately 6
per 1000, and the. 154th town has a notified case rate of ai^proxi-
mately 8 per 1000.

Thus a difference of 41 towns corresponds to a difference of 2 in
the rate, hence a difference of 8 towns corresponds to a difference
of 0-39 in the rate ; therefore the median /•aie=6-39 approximately.

By referring to the original records and writing down, the rate

64 STATISTICS

for each town in the group ' rate 6 and less than 8 ' in which the
median lay, the accurate value of the median turned out to be 6-30.
Theloiver quartile or case rate of the imaginary town, No. J(241),
or 00-25, one-quarter way along the ordered sequence of towns, is
readily shown to be 4-47, and the upper quartile or case rate of
to^v^l No. f (241), or 180-75, is 9-84.
Hence the quartile deviation

= i(9-84-4-47)
=^2-69.

With this may be compared |(S.D.) = |(4-37)=2-92.
Again, the mean deviation measured from 7
=2(^)
= 3-253.

Measured from the mean, it becomes

=:3-253+^'^^^[(41 + 69+39+5)-(29+22+16+7+5+3+4+l)]
241

= 3-253+ (0-564)(67)/241

=3-41

and this may be compared with |(S.D.)=|(4-374)=3-50.

If we estimate the mode by inspection of the frequency graphs in

figs. (2) and (3), we should say it comes between 5 and 6 ; supposing

we call it 5-5, very roughly.

In this case, taking the values actually calculated for mean and

median,

(mean— mode)= 7-56— 5-50

=2-06,

and 3(mean— median)=3(7-56— 6-39)

= 3(1-17)

= 3-51 ;

so that the rule

(mean— mode) = 3(mean— median)

is far from being true according to these results ; this is partly due,
of course, to the very unsymmetrical character of the distribution.

The relative positions of the mean, median, and modal, points
as calculated are indicated in figs. (2) and (3) by three lines drawn
parallel to Oy through these points to meet the graph.

Finally, skewness:^ (mean— mode)/S.D.=2-06/4-37=0-47.

Example 3. — The next example deals ^\'ith the deaths of infants
mider one year, out of every thousand born, in 100 great towns in
the United Kingdom during the thirteen weeks ended 4th Aj)ril 1914.

FREQUENCY DISTRIBUTION

65

The details of the calculation may be left in this case to the reader,
who is recommended to follow the method shown in the last example
so far as possible throughout, including the plotting of the distribu-
tion in different ways. The statistics are as follows : —

Table (17). Death Rate of Infants under 1 Year
PER 1000 Births.

(1) (2) (3) (4)

No. of Towns

No. of Towns

Death Rate.

with Death Rate

Death Rate.

with Death Rate

as in Col. (1).

as in Col. (3).

30 and under 40

1

120 and under 130

16

50 „ 60

3

130 .. 140

11

60 „ 70

2

140 „ 150

10

70 „ 80

6

150 „ 160

8

80 „ 90

7

160 „ 170

3

90 „ 100

6

170 „ 180

1

100 „ 110

11

200 ,., 210

1

110 „ 120

13

240 „ 250

1

1

The more important results are : —
Arithmetic mean=118-9 ; S.D. = 32-2 ;

median^ 120-9 ; quartile deviation=19-5.

Example (4). — As another examjole corresponding details may be
worked out for the following temperature records taken at noon
at a certain spot in Chester week by week during a period of time
covering five years, the results in this case being : —
mean=55-10; S.D.=10-33 ;
median=54-88 ; quartile deviation=7-94

Table (18). 257 Weekly Records of Temperature (Fahrenheit).

(1)

(21

(3)

(4)

Temperature

No. of Records

Temperature

No. of Records

Limits in

between Limits

Limits in

between Limits

Degrees.

shown in Col. (1)

Degrees.

shown in Col. (3)

25-5-29-5

1

53-5-57-5

30-5

29o-33o

1

57-5-61-5

31-5

33-5-37-5

9

61-5-65-5

30

37-5-tlo

11-5

65-5-69-5

26

41 •.5-45-5

28

69-5-73-5

13-5

450-49-5

31-5

73-5-77-5

4

49-5-53-5

36-5

77-5-81-5

3

66 STATISTICS

Before closing the chapter a sHghtly different manner of graphing
the statistics is worth noticing, as it provides us mth a fairly quick
though rough alternative method of determining the mode and
median.

Take, for example, the examination marks data wliich for this
purpose must first be thrown into the second form shown below
Table (7). We mark off on some convenient scale along OX dis-
tances 5, 10, 15, 20 ... 65 from 0 to represent these numbers
of marks respectively, and at the points obtained we erect lines
parallel to OY of lengths 5, 14, 42, 91 . . . 514 to represent the
numbers of candidates who obtained not more than 5, 10, 15, 20
... 65 marks respectively. A freehand curve is then drawn
through the summits of these lines in the manner indicated in
fig. (4), starting from a height 5 and rising to a height 514 above
the axis OX. It is called an ogive curve.

By means of this curve we can approximately state at once how
many candidates obtained any given number of marks or less.
Suppose, for example, we wish to know how many candidates
obtained 22 marks or less, we have only to measure off a distance
22 from 0, represented by ON, and erect a perpendicular NP to
meet the curve at P. Since NP=110 we infer from the manner in
which the curve has been formed that 110 candidates obtained
22 marks or less, so that, incidentally, the 110th candidate from
the bottom must have obtained approximately 22 marks. This
suggests that by worldng backwards we can also read off roughly
the number of marks gained by any particular candidate when his
order in the list is known. Thus, to find the median, i.e. the marks
due to candidate No. 257-5, we merely draw a line parallel to OX
at a height 257-5 above it and the portion of this fine cut off between
the curve and OY measures the median. The value given by this
method is approximately 31-5. Similarly the quartiles are found
by drawing lines parallel to OX at heights 128-5 and 385-5 above
it with results about 23-3 and 39-2 respectively.

Again, as we gradually increase the number of marks, the number
of candidates getting that number of marks or less must increase
also, but the rate of tliis second increase is variable. Th« reader
will perceive that where the height above OX changes slowly the
gradient of the curve is small, but where it changes by big steps
the gradient is steep, and it is at its steepest just in the neighbour-
hood where the greatest addition is being made to the height as
the marks increase, i.e. where the frequency of additional candi-
dates is at its greatest, so determining the mode : this should be

FREQUENCY DISTRIBUTION

67

clear on a comparison of the two arrangements of the data in and
below Table (7). By sliding a straight-edge along the contour of
the curve we can estimate approximately where the curve is
steepest, for at this point the direction of turning of the ruler or

Y

m 1,

..^

m^

J^

/ ---r- -

/

^ ^ "^ '

4.50

r X

\

1

7

Jl

/

unn

ZRC it"

^^g . ._^ Upper -6u<

It tiIe-iL!n& —

P

,,^,_.._,^ ^

-1-

■S

-i 4'^ '

1 . :: :

-!3 "^^U

r

^

F- -f- .. _

s:

i -

^

'

300

?

7

t :

Q>

lA

' . Vledian ..Li^e

"S oi^in "

. _

T t

1

^

,D

^

■ _,

7

4: t

.

7

f

■'

"r

5( 1-

Lo\ rer yuartile Line

^,^__

^ t:

A- -

jt

i \

j±

I

t

1

J

^ 4

J^ ■-

n J

r\ V ^

~ir

10

70

20N 30 40 50 60

Number nf Marks

Fig. (4). Graph showing the Number of Candidates who obtained not
more than any given Number of IMarks.

straight-edge must change. This gives for the mode a value in the
neighbourhood of 32.

It might be advisable to treat the other examples by this method
also, so as to compare results.

aiAPTER VIII

GRAPHS

From the mathematical point of view graphs may be regarded as
the alphabet of Algebraical Geometry.

We can locate a point in a plane, relative to two perpendicular
lines or axes as they are called, OX, OY, which serve as boundaries

of measurement, when we know y and x,
its shortest distances from these boun-
daries. This fact serves to connect up
Geometry, in which points are elements,
with Algebra, in which a:'s and ?/'s,

X standing always for numbers, are ele-
ments. The names abscissa {ah — from,
and scindo — I cut) and ordinate are given to x and y, or, when we
refer to them together, they may be spoken of as the co-ordinates of P.

The celebrated French pliilosopher, Descartes (1596-1650), was
the founder of Cartesian Geometry, and if we may venture to com-
press the essence of his system into a single statement, it is this —
When a point P is free to take up any position in a given plane,
its X and y are quite independent : they may be allotted any values
irrespective of one another. Suppose, however, that P is constrained
to lie somewhere on an assigned
curve, such as APB in the figure,
then X and y are no longer inde-
pendent, for, so soon as x is fixed,
y is fixed also ; it follows that in
this case some relation, algebraical
or otherwise, such as y=x^ — 2x-\-l,
must exist between x and y, and the relation may be called the
equation of the curve which gives rise to it.

Now, if to every curve there corresponds in this way some
equation and to every equation some curve, it seems likely that the
simpler the curve the simpler will be the corresponding equation,
and vice versa. In fact, the student who does not know it already

GRAPHS

69

need only refer to the most elementary treatise on graphs to find
that every equation of the first degree in x and y, i.e. one which does
not involve any x^, y^, xy, or higher powers, represents some straight
line. Any such equation, e.g.

x'-3y+12-0,
can be at once thrown into
either the form

(1)

y

=1,

-12

where — 12 and 4 are intercepts
made by the line on the axes
OX and OY ; or

(2) y=Jx-+4,

where ^, i.e. 1 in 3, is the measure of its gradient and 4 the height
above the origin at which it cuts the axis OY.

Further, every equation of the second degree in x and y, which
may involve x^, y^, and xy, but no higher powers, represents geo-
metrically some conic, a familj^ of curves comprising the parabola,
the ellipse, and the hyperbola, with the circle and two straight
Lines as particular cases. The earth and other planets, likewise
comets, in their journeys through space travel along curves belonging
to the same family, one of ancient and historical connections.

These conies need not, however, detain us, and we pass on at

once to an example of a cubic graph to show how a very little

X knowledge of the theory may be put

to some practical use. Sujipose a
box manufacturer has a large number
of rectangular sheets of cardboard,
3 ft. long by 2 ft. broad, and he
wishes to make open boxes with them
by cutting a square piece of the same
size out of each corner and turning
up the flaps that are left. How big
should the squares be if this is to be
done with as little ysaste as possible ? Clearly this is commercially
an important type of jsroblem to solve.

Let us denote a side of the square to be cut out of each corner
by x feet. Then the bottom of the required box will have dimensions

(3-2.r) ft. by (2-2x) ft.

and its depth will be x ft.

3-2. V

[The

3ft. >-

ihaded flaps are bent upwards
along the dotted lines.]

70

STATISTICS

Hence the capacity of the box when completed will be

x{3—2x){2-2x) cu. ft.,

and he makes best use of the material who jsroduces the most
capacious box. Call this expression y and let us find the values
of y corresponding to different values of x so as to be able to draw
roughly the curve of which the equation is

y^x{3-2x){2-2x)

(1)

Table (19). Table of Corresponding Values of x and y
IN THE CcjRVE y=x{3—2x){2—2x).

X

2x

(3-2x)

(2-2.r)

a;{3-2.c)(2-2a;)

y

-1

-2

5

4

-20

-20

-h

-1

4

3

- 6

- 6

-I

-*

h

IF

- 219

0

0

3

2

0

0

+ i

+ i

i

■^

+\%

+ 0-94

+ i

+ 1

2

i

+ 1

+ 1

H

+ ij

3

2

*

+ A

+ 0-56

+ 1

+ 2

1

6

0

0

+ U

+ #

h

-A

-h

- 0-31

+ ii

+ 3

0

-i

0

0

+ 2

+ 4

-1

-2

+ 4

+ 4

+ 2^
0-2

+ 5

-2

-3

+ 15

+ 15

0-4

2-6

1-6

(0.2)(2-6)(1.6)

0-83

0-4

0-8

2-2

1-2

(0.4)(2-2)(1.2)

106

0-6

1-2

1-8

0-8

(0-6)(l-8)(0-8)

0-86

0-8
0-38

1-6

1-4

0-4

(0.8)(1.4)(0.4)

0-45

0-76

2-24

1-24

(0-38)(2-24)(l-24)

1055

0-39

0-78

2-22

1-22

(0-39)(2-22)(l-22)

1-056

0-40

0-80

2.20

1-20

(0-40)(2-20)(l-20)

1-056

0-41

0-82

2-18

1-18

(0-41)(2-18)(M8)

1-055

We get a tolerably good idea of the shape of the curve by plotting
the points {x, y) shown in Table (19) from a:= — f to a:=+2 as in
fig. (5). It is simply a matter of practice to be able to determine
the whole curve from a few points in this way, and the greater the
number of points plotted the more accurately will it be possible
to draw the curve. It should be noticed that the points for which
?/=0 are in a sense key- points to the curve : they are readily

GRAPHS

71

-6

0 0-25 0-50 ■ 0-75

Length of Side of Square cut out

Fig. (5).

100 X

72 STATISTICS

found by maldng the factors separately zero in the right-hand side
of equation (1), namely x=0, 3— 2a:=0, and 2— 2a;=0, and by
jjlottuig them first they serve as a guide to the jjosition of points
subsequently plotted.

We want to Ivnow for what value of x the capacity of the box, y,
is greatest and the preliminary plotting is enough to indicate a
maximum value for y between x=0 and x=l, for the curve first
rises and then falls between these two limits. In order to discover
more exactly where the maximum is located we therefore plot
in addition the points corresponding to a;=0-2, 0-4, 0-6, 0-8 respec-
tively, and this is done on a larger scale than that used in the
first diagram because the accuracy is thereby increased (see fig. (5)
inset).

The calculations and figure suggest that the maximum required
is very near the point for which a;=0-4, so we next work out values
of y in this neighbourhood, corresponding, say, to a:=0-38, 0-39,
0-40, 0-41, with the results shown at the foot of Table (19). From
these we conclude that to a fair degree of accuracy the maximum
value of y is given by taking a:=0-395. It would be possible in
the same way to calculate more decimal places, but we have gone
far enough to make the method clear.

Hence the side of each square cut out should be of length

0-395 ft., or 4| in.

Whenever the value of one variable, y, dej)ends ujDon that of
another variable, x, in such a way that when x is given y is kno'WTi,
so that y may be termed a function of x, corresponding values of
x and y can be plotted — as was done in the example just discussed —
and a curve drawn by joining up the points obtained, the relation
which connects x and y being the equation of this curve. More-
over, it is possible, by calculating enough points from the equation
and plotting them, to get the curve as accurately as we please.

In Statistics, however, we usually have to start the other way
round and reach the equation, if at all, last. We make observations
of two sets of variables, a set of cc's, and a set of y's, one of which
is dependent in some way upon the other — e.g. y, the dependent
variable, might denote the number of individuals observed to have
a certain organ of length x, the independent variable — and thus
we get pairs of corresponding values like {x^, y^), {x^, y^), {^z, Vz) ■ • ■
We met with examples of this method of recording results in the
last chapter, and we need only rej^eat here that its chief virtue is
suggested in the root of the word itself — it is more graphic than a

GRAPHS 73

long table of figures and, by means of it, many of the essential
features of a problem are immediately seized upon.

Now for some j^urposcs it may be necessary to go further and
to find what curve would best fit the points plotted, assuming they
were numerous enough, and what equation between x and ?/ would
best describe the curve. But the graj)hs we meet in Statistics,
bearing, for instance, uj)on sociological or biological problems, are
in general much more wayward than the mathematical kind we
have referred to in the present chapter : it is impossible to set
down simple equations to which they can be rigidly confined, and
when we are unable to find any relation which accurately and
uniquely defines ?/ as a function of x we must rest satisfied with the
most manageable equation and the best fit we can get.

In sciences such as Engineering and Physics it is often possible
to fix upon two mutually dependent variables, x and y, and to
observe enough corresponding values of each to enable us to draw
a graph which answers very closely to the true relationship between
them, so that a connecting equation can be determined ; e.g. we
may plot the amount of elastic stretch, y, in a wire Avhen diiierent
weights, a-, are hung from the end of it, and it is found that y is
directly proportional to x. If we deal in this way with some
simple figures which are amenable to our purj)ose it may help to
make clear the nature of the same problem in Statistics.

The following corresponding values of x and y were given in a
Board of Education Examination (1911) :—

a;=100, 1-50, 2-00, 2-30, 2-50, 2-70, 2-80 ;
y=0-n, 1-05, 1-50, 1-77, 203, 2-25, 2-42.

Allowing for errors 'of observation, it was desired to test if there
was a relation between y and x of the type

y^a+bx^- . . . (1)

In the first place, the shape of the curve obtained by plotting
y against x, as in fig. (6), would, to the initiated, probably suggest
a parabola, the equation of which is of type (1). In order to test
its suitability we proceed to plot y against x^, or, putting x^=^, we
plot y against |. If equation (1) holds, then, in that case

y=a^bi . . . (2)

should also hold, and this, in (^, y) co-ordinates, represents a straight
line. The result of plotting y against | should therefore be a
number of points approximately in a straight line — we say ' ap-
proximately ' to allow for errors of observation in the original data.

74

STATISTICS

Now from the given statistics corresponding values of ^ and y
are, since ^—x- : —

1=1-00, 2-25, 4-00, 5-29, 6-25, 7-29, 7-84 ;
y=0-n, 1-05, 1-50, 1-77, 2-03, 2-25, 2-42 ;

Y

~

2-5

1

/

J

J

{

20

f

/

/

i

/

'

1-5

r

/

/

/

10

;

/*

^

<»

.^

-

0-5

-

r>

_

__

_

_

_

^

_

^

H

_

_

i_

0-5

10

1-5 20

Fio. (6),

2-5

30

and the resulting grajih, fig. (7), is very approximately a straight
line. To determine its equation, choose two points (not too close
together) on the line, which has been drawn so as to run as fairly
as possible through the middle of the points plotted, and, in choosing,
take points which lie at the intersections of horizontal and vertical
cross lines (the printed Unes of the graph paper) if such can be
Y

4 5

F.o. (7).

i

found, because their a;'s and 2/'s can be read off with ease and
accuracy. Two such points are

(2-8, •1-2) and (6-0, 2-0),

GRAPHS 76

and since each of these points lies on the Hne whose equation is

we have

Subtracting, we get

Therefore

l-2=a+6(2-8)
2-0=a+6(6-0).

0-8=6(3-2).
b=l

Hence a=2— f=J.

Thus the equation of the hne is

i.e. 4y=|+2,

and the law coiuiecting x and y is therefore

4y=x^-{-2.

The follo\\ing statistics, the result of an experiment in Physics
to verify Boyle's Law, may be treated in the same way. .r is a
number proportional to the volume of a constant weight of gas in a
closed space, and yis a, number proportional to its absolute pressure.
Corresponding values of x and y observed were : —

X—

46-89

41-9G

40-33

38-88

37-37

36-06 34-71 33-47

y^

76-32

85-38

88-93

92-36

96-09

99-61 103-51 107-51

^x= 32-39 31-08 29-97 28-76 27-26 25-32 24-04
[?/=lll-09 115-69 120-05 125-08 131-99 14209 149-81.

Boyle's Law states that the product xy is constant, and this may be
tested by putting |=- and plotting y against | ; the jjoints obtained

X

should be approximately in a straight hne.

Now in Statistics, as we have already explained, the exact con-
nection between the variables, x and y, is rarely so clear, though
the absence of law is not so complete as it might seem at first sight.
At this stage, however, we need not enter into the difficult question
of curve fitting : if draA\Ti M-ith care and used with judgment much
that is of value may be learnt by simple plotting and by connecting
up the resulting points by straight lines or a freehand curve. We
shall briefly explain or illustrate by examples how graphs and

76 STATISTICS

graphical ideas may be- used to serve three distinct purposes,
namely : —

(1) to suggest correlation or comiection between two different

factors or events ;

(2) to supply a basis for finding by interpolation some values of a

variable when others are known ;

(3) -AS pictorial arguments appealing to the reason through the eye.

We reserve (2) and (3) for the next chapter and proceed at present
with an example of (1).

Correlation suggested by Graphical means. Consider the index
numbers, col. (2) Table (20), showing the variation from year to
year in wholesale prices between the years 1871 and 1912. It is
not an easy matter to take in satisfactorily the meaning of such a
mass of bare figures, but they are much easier to grasp when plotted
in a graph.

In this case the numbers x, representing years, and the numbers y,
representing prices, are measures of things of quite a different char-
acter, so that it is not necessary to take the x and y units of the
same size. Moreover they need not, in a case of this kind, neces-
sarily vanish at the origin, but it is convenient to draw the graph
in such a way that it shall occupy the greater part of the space at
our disposal. Thus, we have roughly 80 small squares across the
breadth of our graph paj)er, and between 1871 and 1912 we have
roughly 40 years ; we therefore take two sides of a square to 1 year
and mark off the years 1870, 1875, 1880, . . ., along an axis or
base line parallel to the breadth of the paper, as shown in fig (8).
Again we have roughly 70 small squares in the available space
from this base line to the top of our graph paper, and the whole-
sale price index numbers vary from 88-2 to 151-9, a range of 63-7 ;
we therefore take one side of a square to correspond to a difference
of 1 in the price index number, and mark off the prices 90, 100,
110, ... , along an axis parallel to the length of the paper, as
shown in the figure.

We then plot points to represent the numbers in col. (2) of
Table (20). Thus, in 1880 wholesale prices stood at 129 ; we there-
fore travel along the width of the paper till we reach 1880 and
then upwards until we are opposite the 129 level on the axis of
prices, inserting a dot to mark the position. Similarly for all other
points, and the required graph is given by joining them up in
succession.

GRAPHS

77

Table (20). Marriage Rate and Wholesale Prices
Index Numbers.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Nine Years'

Difference be-

Nine Years'

Difference be-

Year.

Prices.

Average

tween Nos. in

Marriage

Average of

tween Nos. in

of Prices.

Cols. (2) & (3).

rate.

Marriage rate.

Cols. (5) &(()).

1871

135-6

167

1872

145-2

174

1873

151-9

176

1874

146-9

170

. .

1875

140-4

139-3

+ M

167

164

+ 3

1876

137-1

138-6

-1-5

165

162

+ 3

1877

140-4

136-5

+ 3-9

157

159

- 2

1878

1311

133-8

-2-7

152

157

- 5

1879

125-0

131-5

-6-5

144

155

-11

1880

129-0

128-5

+ 0-5

149

153

- 4

1881

126-6

125-2

+ 1-4

151

151

1882

127-7

120-8

+ 6-9

155

149

+' 6

1883

125-9

117-2

+ 8-7

155

148

+ 7

1884

1141

114-7

-0-6

151

148

+ 3

1885

107-0

111-8

-4-8

145

149

- 4

1886

101-0

109-2

-8-2

142

149

- 7

1887

98-8

106-9

-8-1 I

144

149

- 5

1888

101-8

104-2

-2-4

144

149

- 5

1889

103-4

102-5

+0-9

150

149

+ 1

1890

103-3

101-0

+2-3

155

149

+ 6

1891

106-9

99-9

+ 7-0

156

150

+ 6

1892

101-1

98-7

+ 2-4

154

151

+ 3

1893

99-4

97-4

+ 2-0

147

153

- 6

1894

93-5

96-3

-2-8

150

155

- 5

1895

90-7

95-0

-4-3

150

156

- 6

1896

88-2

94-3

-61

157

156

+ 1

1897

90-1

93-8

-3-7

160

157

+ 3

1898

93-2

93-4

-0-2

162

158

+ 4

1899

92-2

93-8

-1-6

165

159

+ 6

1900

100-0

94-7

+ 5-3

160

159

+ 1

1901

96-7

95-7

+ 1-0

159

159

1902

96-4

96-9

-0-5

159

158

+ 1

1903

96-9

98-3

-1-4

157

158

- 1

1904

98-2

99-5

-1-3

153

156

- 3

1905

97-6

100-0

-2-4

153

155

- 2

1906

100-8

101-3

-0-5

157

154

+ 3

1907

1060

102-8

+ 3-2

159

153

+ 6

1908

1030

104-8

-1-8

151

153

- 2

1909

104-1

. .

147

. .

1910

108-8

150

1911

109-4

152

1912

114-9

155

78 STATISTICS

It is comparatively easy from this graph to trace the change
in prices from year to year and from decade to decade : for example,
we note that from 1873 to 1896 the tendency of prices was on the
whole doAvnward, and from 1896 to 1910 the tendency was upward.
Also on the assumption — not necessarily valid — that prices have
varied continuously, or at least consistently, during the intervals
between the dates to which the records refer, it is possible to read
off intermediate values from the graph : e.g. midway between 1883
and 1884 we get the figure 120 as the index number for prices.

On the same graph sheet we have also plotted the marriage rate
from year to year during the same period. The numbers are given
in col. (5) of Table (20). This rate varies from 142 to 176, a range
of 34, and we have a range of 40 small squares at our disposal in
plotting ; a difference of 1 in the marriage rate has therefore been
taken to correspond to one side of a square, and the marriage rates
140, 150, 160 .. . are accordingly marked along the axis perpen-
dicular to the same base line as before, which is used again to
measure the passage of years, but the second graph is drawn below
the line whereas the first was drawn above it. In this way we
are able to compare the two graphs, namely, the one registering
the change in prices and the one registering the change in marriage
rate from year to year.

It is interesting to observe that the two seem to be not uncon-
nected : they go up and down almost in the same time, and moun-
tains and valleys in the one correspond roughly to mountains and
valleys in the other ; in other words, there is some kind of correlation
or reciprocal relation between them. Now these mountains and
valleys are largely the result of what may be caUed short-time
fluctuations, and it is important to distinguish between these changes
which are transient and the more permanent or long-time changes.
In order to get rid of the former, which sometimes conceal the
latter, the following device has been adopted : noticing that the
wave period, the length of time taken for each complete up-and-
dowTi motion, is one of about nine years, nine-yearly averages have
been taken of the figures for wholesale prices right down col. (2)
of Table (20) ; thus 139-3 is the average of the index numbers from

1871 to 1879 inclusive, 138-6 is the average of the numbers from

1872 to 1880 inclusive, and so on, the results being recorded in
col. (3). When the points corresponding to these numbers are
plotted we get the broken line in fig. (8) passing through the body
of the original graph of prices and indicating its general trend in
the course of years as separated from the temporary fluctuations.

GRAPHS

79

1870 1875 1880 1885 189c 1895 1900 1905 I9I0

Fio. (8). Graph showing Variation in "Wholesale Prices Index Numbers.

j^j 4 1'^-^ 3^ ^^j ^7^\

1870 1875 i88p 1885 1890 1895 1900 1905 1910

Fio. (9). Graph showing Variation in ]\Iarriage Rate Index Numbers.

80 STATISTICS

The same procedure has been followed with the marriage rate
statistics ; the nine-yearly averages are showTi in col. (6) of Table (20),
and their graph appears as a broken line passing through the body
of the original marriage rate graph in fig. (9).

G'ra

)h 1 sHowin'g

Fluctuations from theirlNine-yearlv AveraWe

5

^>

/

0

ft

h^Iiid

ex Numbers

of

Wholesale Prices

3

/

«]

«

1'

\

I

P+5

/

I

\

•

1

•^

i

1

/

I

(

^>

J

E

5 <

J

J

I

/

i

/

"? n

V

/

n

r

\

\

/

S i8;

\

/

i(

8o

18

8.S

1

1890

\

189.

3

V

'15

00

N

»v,

19

3S

19

10

s

\

[

\

1

N

\

,,

«]

1

\

'

\

\

"C

\

\

\

1

•S-5

OS

\

:

\

/

V

,

\/

s

V

I

/

1

it^

r— <

^^-10

+10

91

^

u

=)

/

I

1+5

\

^

/

1

\

\

I /

1

i

S

1

1

/

\

1/

1

\

.fc

1

/

1

1

o

\

/

I

\

? 0

\

1

1

s

/

V

\

§> '8:

s

\

i8

80/

ii

88^ 1

J

18

qo

1

i8q<

i<!

00

\

10

1

IQ

10

o

/

1

I

1

\

r

1

1

\

\

J

§

1

1

\

i

c

k

1

^ '

\

/

/

\

lu

\

/

Gr£

iph

sh

OW

in

eP

uctiia

;io

ns

fr

om

the

ir Nin

e- ye

ar

y

Av

er

ag

;s

of Mar

nag:

1-rates | |

1

1

Tl 1 i 1 1

1

_J

Fig. (10).

Suppose we wish on the other hand to study the short-time
fluctuations as distinct from the ' secular trend,' we may do so
by forming the differences between the numbers for each year
and the corresponding nine-yearly averages, and plotting these
differences on convenient scales.

The numbers obtained in this way are recorded, -ndth their proper
signs — positive if above the average, negative if below — in cols. (4)
and (7) of Table (20), and the grajDhs of these differences are drawn,

GRAPHS 8i

one below the other for comparison, on the same graph sheet
(fig. 10). The agreement in fluctuation from the average between
the two factors, marriage rate and prices, is more easily remarked
now than it was in the original graphs. High prices go as a rule
hand-in-hand "with prosperous times, and such times lead to more
frequent marriages. This statement must not be taken to imply
that when prices are high the times are always necessarily pros-
perous for the community as a whole : the lie direct would be given
to such an impHcation by any one who had experienced abnormal
war conditions.

After about 1892, while the fluctuations continue to be similar,
a tendency appears for the marriage rate graph to reach each
extreme point about a year in advance of the other, as though an
increase in marriages raised prices and a decrease lowered them.
There is no doubt that any economic change, especially if it takes
place on a large scale, will set up a system of corresponding forces,
sometimes in unexpected directions, actions and reactions succeed-
ing one another at intervals hke tidal waves producing each a back-
wash as it breaks, but such effects, even when anticipated in theory,
are not always easy to unravel in practice.

The comparison we have been discussing between changes in
prices and marriages is suggested in Sir W. H. Beveridge's Unemj)loy-
ment. The whole book A^dll repay careful study, but it contains
one particularly illuminating chapter on ' Cyclical Fluctuation ' with
a chart labelled ' The Pulse of the Nation,' because of the remark-
able picture it gives of the ebb and flow of the tide of national
prosperity. It consists of a series of curves representing respec-
tively : —

(1) bank rate of discount per cent. ;

(2) foreign trade as measured by imports and exports per head

of the population ;

(3) percentage of trade union members not returned as unem-

ployed ;

(4) number of marriages per 1000 of the population ;

(5) number of indoor paupers per 1000 of the population ;

(6) gallons of beer consumed per head of the population ;

(7) nominal capital of new companies registered in pounds per

head of the population.

The interesting thing about these curves is to see the way in
which they move in waves of varying size up and down almost
together, showing a connection between such phenomena more

F

82

STATISTICS

intimate than one might at first have suspected. A note of caution
must be inserted here liowever : causal comiection must not be too
confidently inferred in discussing the correlation of characters
changing simultaneously with time ; because two events happen
together, one is not necessarily caused by the other.

An instructive article bearing on this point ajjpeared recently in a
periodical well laiown to students of social problems. It was there
stated that high positive correlation exists betA^een birth rate and
infantile death rate : in general the two rise or fall together, whence
Neo-Malthusians argue that the Avay to lower a death rate is to
lower the birth rate. The writer then contrasts Bradford, the last
word in the scientific care of infants, with Roscommon, where con-
ditions as to wealth and child welfare are the very reverse, and
points out that Bradford has a birth rate of 13 and an infant death
rate of 135, while Roscommon has a birth rate of 45 and an infant
death rate of 35. These figures, he suggests, prove instantaneously
that the Neo-Malthusians are guilty of the commonest of all fallacies,
they confound correlation with causation.

As an exercise in plotting the reader may see whether he can
discover any suggestion of correlation between crime and unem-
ployment by comparing the follo\\T.ng statistics, showing the number
of indictable offences tried in the United Kingdom and the trade
union unemj)loyed percentages respectively from 1861 to 1905 : —

Table (21). Number of tried Indictable Offences and
Trade Union Unemployed Percentages (1861-1905).

No. of Indictable

Trade Union

1

No. of Indictable

Trade Union

Year.

Offences tried

Unemployed

Year.

Offence-s tried

Unemployed

(in thousands).

percentages.

(in thousands).

percentages.

1861

560

3-7

1874

53-5

1-7

1862

61-3

60

1875

500

2-4

1863

61-4

4-7

1864

58-4

1-9

1876

519 •

3-7

1865

59-9

1-8

1877

53-8

4-7

1866
1867

57-6
59-5

2-6
6-3

1878
1879

560
550

6-8
11-4

1868

62-4

6-7

1880

60-7

5-5

1869

61-3

5-9

1881

60-6

3-5

1870

561

3-7

1882

63-3

2-3

1871

531

1-6

1883

60-8

2-6

1872

51-9

0-9

1884

59-6

81

1873

53-5

1-2

1885

56-4

9-3

GRAPHS

83

Table (21). Number of tried Indictable Offences and Trade
Union Unemployed Percentages (1861-1905) — Continued.

Year.

No. of Indictable
Offences tried
(in thousands).

Trade Union
Uneinplo^-ed
percentages.

Year.

No. of Indictable
Offences tried
(in thousands).

Trade Union
Unemployed
percentages.

1886

56-2

10-2

1896

50-7

3-3

1887

56-2

7-6

1897

50-7

3-3

1888

58-5

4-9

1898

52-5

2-8

1889

57-6

2-1

1899

50-5

20

1890

550

21

1900

53-6

2-5

1891

541

3-5

1901

55-5

3-3

1892

58-3

6-3

1902

571

40

1893

57-4

7-5

1903

58-4

4-7

1894

56-3

6-9

1904

600

60

1895

50-8

5-8

1905

61-5

5-0

The chief point of difficulty in plotting such graphs is the initial
one of fixing upon the most convenient scales to use, and in this
matter hints only can be given, faciUty will come by practice. An
examination of Table (21) shoMs that the data cover a period of forty-
five years which can be marked off horizontally along a base line so
as just to fit comfortably into the available space across the graph
paper. The unemi^loyed percentages vary between 0-9 and 11-4,
giving a range of 10-5. Similarly the indictable offences recorded
(in thousands) present a range of 13-3. We might therefore very
well choose the same vertical scale for the measurement of indict-
able offences and unemployment, but, in order that the graphs
may run more or less together (without exactly overlapjDing) for
the sake of comparison, only the unemployment zero need be taken
actually on the base line, whereas the indictable offences may have,
say, the number 50 (thousand) at that level ; also it will be con-
venient to show the scale for unemployment on the right side
and the scale for offences on the left side of the paper.

An example dealing with matters somewhat different is provided
by a comparison of changes from week to week in —

(1) the mean air temperature ;

(2) the percentage of possible sunshine ; and

(3) the rainfall.

The following is a record of observations taken at Greenwich in
1912 [data from London Statistics, vol. xxiii.] : — ■

84

STATISTICS

Table (22). Weekly Meteorological Observations
at gree^'wich (1912).

Mean Air

1

1

Mean Air

Week

Tempera-
ture—

Per-
centage of

Rainfall

1

Week

Tempera-
ture—

Per-
centage of

Rainfall

ended —

Degrees

possible

in
inches.

ended —

Degrees

possible

in
inches.

Fahren-

Sunshine.

i

Fahren-

Svinshine.

heit.

1
1

heit.

Jan. 6

45-7

7

0-76

' July 6

58-7

15

0-36

13

41-9

15

0-45

13

670

46

0-20

20

40-2

1

0-93

20

65-8

44

004

27

38-9

8

0-88

27

64-8

31

016

Feb. 3

300

21

002

Aug. 3

57-8

33

0-54

10

39-5

15

0-52

10

57-6

28

1-26

17

45-5

11

0-44

17

56-2

14

0-23

24

47-4

6

0-65

24

57-2

24

1-27

Mar. 2

49-8

21

0-52

31

56-9

27

1-33

9

44-6

31

0-79

Sept. 7

54-8

36

0-21

16

451

16

019

14

52-4

14

002

23

42-7

15

108

21

53-6

22

000

30

510

46

005

28

51-5

59

002

Apr. 6

48-0

43

007

Oct. 5

48-8

36

2-30

13

45-6

43

002

12

460

53

000

20

500

50

000

19

49-8

38

013

27

52-6

76

000

26

45-4

23

0-88

May 4

501

32

0-21

Nov. 2

491

31

0-55

11

59-7

29

006

9

47-2

6

018

18

55-2

49

0-69

16

43-3

3

017

25

541

38

019

23

46-2

6

0-31

June 1

570

47

017

30

40-4

13

106

8

54-2

35

0-99

Dec. 7

42-4

9

0-31

15

58-1

48

0-39

14

490

2

0-62

22

61-7

56

0-65

21

44-4

19

0-59

29

60-2

45

0-30

28

48-1

8

1-22

The rainfall graph here should be dra^^-n reversed {i.e. so that
it goes up as the rainfall goes down in amount, and vice versa),
because one would expect in general much rain to go with httle
sun and low temperature.

The range of temperature during the year is 37 degrees, of sun-
shine 75 per cent., and of rainfall 2-30 in. Hence the vertical
scales for these three graphs might be chosen so that, roughly,
40 units of temperature should correspond to 80 units of sunshine
and 2 units of rainfaU. Also the zeros of the three variables should
be so placed, relative to the horizontal base line registering the
weeks, that the three graphs may be conveniently compared -n ithout
causing confusion by too closely overlapping.

CHAPTER IX

GRAPHS {continued)

Graphical Ideas as a Basis for Interpolation. It frequently happens
in statistical records that aw kward gaps occur which recpiire to be
filled in ; this may be due to the fact that no record has been
made, or that it has been made with insufficient detail, or that it
has been lost or destroyed. Cases in point arise in connection with
returns like that of the Census which can only be undertaken every
few years, so that if figures are wanted for any intervening year,
as they are in very many instances, an estimate has to be made
from the knoAvn results of the years recorded. It is imperative, for
example, for many purj^oses of local or national government, to
be able to find with a fair degree of accuracy the population of
county boroughs and urban or rural districts at any given time,
to know the number of workers engaged in different occupations,
the amount of land in pasture and under various crops, the con-
dition of the people as to housing, of the children as to education,
and so on indefinitely.

Symbolically, with the same notation as we have used before,
we conceive the statistics in tabular form, like

Ui, y-i, Vs ' • • Vn

each y denoting the frequency corresponding to the character
measured by its companion x, e.g. the x's may stand for successive
dates and the y's for the frequencies of the population of a certain
district at those dates. If it happens that one or more of the y's, in
between the first and the last recorded, are missing, the problem is
to estimate the missing values by some method of interpolation, as
it is called. Various methods of arriving at such estimates are used,
but we shall only refer to the more elementary here.

A rough way of making the estimate, but one which is often as
accurate as the data will allo\^', is to plot the observations, each
(.r, y) being represented bj' a point, and connect them up, if there

85

86

STATISTICS

be enough of them, by a smooth curve drawn freehand P^ Pg P3 . . . P„

[see fig. (11)] ; to find the y proper to any other x we have then
only to draw the ordinate through the point {x, o) and measure the
y at the point where it cuts the curve. This is a not unreasonable

principle to follow, for in effect it
gives due weight to each of the
observations actually recorded,
and it assumes an even course
from each one to the next — a
justifiable assumption in the
absence of any evidence that
some sudden discontinuity of
value has taken place.

If only two observations are
given, represented by the points

Pi {^v Vi) 'int^ P2 (•^2> ?/2)> the
curve connecting them is a straight line, and the y corresponding to
any other x is at once given geometrically, as fig. (12) shows, by

PM PiM

P2M2

P1M2

i.e.

or

2/2 Vi •"^2 •*'i

y-z-Vi

y=yi+'

tl/n JU-t

{X-X^),

the familiar proportional relation which is employed in this simple
case.

F

3^__,„-^-^

F>^ "^

M

y.

V

Example. — Given
Required log 5-826736.

Fig. (12).

log 5-82673= 0-7654249,
log 5-82674=0-7654257.

GRAPHS 87

Here a:,=5-826730 ?/i=0-7654249

^2=5-826740 ?/^=0-7654257

a;=5-826736.

Therefore, by means of the above relation,

0-000010
=0-7654249+0-00000048
=0-7654254.

The logarithmic curve y—logx is, of com-se, not a straight line,
and the value obtained for y only represents a first approximation
to the true value.

When more than two points are given there is bound to be a
margin of inaccuracy, more or less according to the data, intro-
duced in drawing the curve. For an example of this method the
reader may refer back to the curve on p. 67, which was used to
determine the median and quartiles. We may, as we saw, read
off from it the number of candidates who obtained not more than
any stated number of marks : e.g. 300 candidates obtained not
more than 34 marks ; or we may use it the other way round and
find the number of marks obtained by a stated number of candi-
dates : e.g. 10 per cent, of the candidates got less than 17 marks.
Such examples might be multipHed endlessly, and the method will
be found extremely useful when a liigli degree of accuracy is not
looked for. But greater confidence will be felt perhaps in such
results — though the foundation for it may be no more secure in many
cases — if we can translate them from geometrical to algebraical
form, if we can find, that is to say, some formula, like the simj)le
proportional relation already introduced above, which will give
one y when others are knoAMi.

In order to make the argument as general as possible we shall
speak of x and y as variables, and we shall think of the value of y
as depending upon that of x in such a way that when x is given,
y is known or it can be estimated * (in the sense that when the
year is given the population is known or can be estimated).

Suppose

t/=Co+qa:+C2.r2-J- .

[* This is equivalent to assuming that y is some function of .r, say y=J\x), and
clearly some such assumption is necessary if any estimate from the known values
to the unknown is to be possible. Further, for simplicity we assume J\.c) can
be expanded in a Maclaurin's converging series of ascending powers of x, which
simply means that we take the relation between x and y to be of the form
adopted above.]

88 STATISTICS

where the c's are constants to be determined, and their number
can be made to depend upon the number of knowTi values of y
which are used in the estimate.
Geometrically, the equation

2/=Co+Cia:+C2a;2+ . . . +c„a;"

represents a curve called a parabola of the nth. order, and such
a curve could be employed (and uniquely found — there is only one
parabola of the Idnd which will go through all the points) if we
based our estimate upon a knowledge of {n-\-\) y's corresponding
to given .r's, for we could readily make it pass through the (w-f 1)
known points {x^, Vq), [x^, y^), {x.^, y.^, . . . {x^, ?/„) by choosing
the (w+1) c's so as to satisfy the (/? + !) simple linear relations : —

2/i=Co+Ci.ri+C2a%2^ . . . +c„a:i«

2/2 '^^o'T ^1*^2 "rC2'^2 "r • • • ~rCn'^2"

7/„=Co-(-Cj^a:„-)-C2.T„ -(- . . . -j-c„.T„".

When the curve is determined, in other words when the c's are
known, we can find any other y required by substituting the corre-
sponding x in the equation

2/=Co+Cia;+Coa;24- . , . +c„a;«,

i.e. by supposing this point {x, y) to lie on the same curve that goes
through the known jDoints.

It is well to mention here that the parabola is by no means alwaj^s
the best curve for fitting any given statistics, and when the number
of observations is adequate it is possible often to make a more
satisfactory choice. Once the equation of a suitable curve has
been determined the subsequent interpolation or calculation of y
for any given x is not as a rule a very difficult matter. The larger
question of curve fitting in general is reserved for a later chapter.

Example of First Metlwd {fitting with a parabolic curve). Let us
illustrate this process of interpolation by fitting a parabolic curve
to the folloA\ing figures, extracted from Porter's The Progress of
the Nation, giving the annual cost of Poor Relief (excluding insane
and casual) at five-yearly intervals, but with the amount for the
year 1845 omitted : —

Year . . . 1835, 1840, 1845, 1850, 1855]

Cost in £1000 . . . 5526, 4577, ? 5395, 5890/

GRAPHS 89

Assuming that no extraordinary conditions prevailed in 1845 to
cause abnormality in expenditure, let us estimate what the figure
would be for that year judging from the given records just before
and after. Since there are four known points in this case, we take
as the curve through them a parabola of the 3rd order, namely : — •

y=c^j^c^x+c.^x-+c^x^ ; . . . (1)

the four known points wUl then just suffice to determine uniquely
the four arbitrary constants Cq, c^, c^, Cg. Also, since the x class-
intervals are equal, it will simplify the algebra if we measure from
the year 1845 as origin, taking five years as unit for x and £1000
as unit for //, so that we get

x=—2, -1, 0, +1, +2 \
y=552G, 4577, ^/o- ^395, 5890J

where y^ is the number to be determined.

Since all five points are to lie on the curve with equation as in
(1), we have by substituting in that equation —

■8c,

5526-
4577=

=Co-

-2ci+4c2

2/0=

= Co

5395=Co+Ci+C2+C3

5890=Co+2ci+4co+8c3.

Adding the first and last of these equations,

2co+8c2 =5526+5890 .... (2)

Adding the second and last but one,

2co+2c2=4577+5395
or 8co+8c2= 4(4577 + 5395) . . . (3)

Subtracting (2) from (3),

6co=4(4577+5395)- (5526+5890) . . (4)
=4(9972)-(11416)
= 39888-11416
= 28472.
Therefore yo=Co=£4,745,000.

If we only wish to make use of the records for the years 1840
and 1850, the appropriate fitting curve reduces to a straight line

y=Co+Ci^x,

90 STATISTICS

on which we assume the points

(-1,4577), (0,^0). (+1,5395)
to He, so that

4577=Co— Ci

5395=C(,+Ci.
Therefore, adding the first and last of these equations,

2co=4577+5395,
so that yo=Co=£4,986,000.

* Second Method {using a formula connecting the ordinates). When,
as above, the steps from each x to the next are equal, as commonly
happens in practice, it is possible to write doT^n a simple relation
between the y's, kno^vn and unloiown, without introducing the c's
at all. At bottom the method is the same as the last, inasmuch as
the elimination of the c constants by the first method really results
in the same formula for the unkno^\Ti y.

Let us represent the given statistics in this case by

. XQ-^nh\
Vn J

so that, if the fitting curve be

y=Co+Ci.r+C2Ct;24- , . . +c„X'",

we have, by substituting the co-ordinates of the first two points
in this equation,

2/l = Co+Ci(.To + ^) + C2(.l-o + /0"+ . • . +Cn(«0 + ^)''

and 2/o=Co+Ci x^ -^H V + - ' • +c„a-o«.

Hence

2/i-2/o=Ci;i+C2(2.ro?i+A2)_|_ , . , _|_c^(w.To"-iA+ . . .)•

Now this result, which we call the \st differ ence between the y's,
is of (w— l)th degree in Xq, so that by subtracting two of the y's
we have reduced the degree in x^ by 1. Similarly,

2/2-2/i=Ci/i+C2(2.roA+3/i2)+ .... +c„(n.To"-^/^+ . • .)•

Thus we get a series of \st differences, each with the highest
term of the {n—\)th degree in Xq. Treating them as a series of new

[* The non-mathematical reader will do well to omit the rest of this section on
interpolation. ]

GRAPHS

91

ordinates and forming their differences in the same way, we get
what maj'^ be called the 2nd differences between the y's, a series
of ordinates each with the highest term of degree {n—2) in Xq.
Proceeding in this way the Zrd differences between the y's are a
series of ordinates of degree (w— 3) in Xq, the Aih differences are of
degree (w— 4), and so on, until ultimately we reach the nth differences,
which are of zero degree in Xq, and consequently involve only h.
It follows that the ni\\ differences must all be equal in value and
therefore, if we go one steja further and write down the {n-\-\)th
differences, these must vanish altogether.

If the reader finds any difficulty in following the argument
he should test it step by step for himself in the simple case of a
parabola of the third order when it should be perfectly clear.

The formation of the successive differences is conveniently sho\ATi
in Table (23).

Table (23).

Successive Differences of

Ordinates.

First

Second

Tiiird

Fourth

Fiftli

V

difterence

difference

ditlerence

difference

difference

y.-\

A

A 2.

A^.

A4.

A5.

(

yi-y»)

vx)

\

V2-2.'/]+3/o~|
2/3-22/2+2/1/

3/2 - 2/1 )

ys- 32/2+ 33/1 -2/0 'I
2/4 -32/3 +32/2 -3/1 J

Vl

2/4-43/3 + 62/2-^2/1 + 2/0)

2/3 - 2/i!

2/5 - 53/4+IOV3 - 10!/2+5;/i - 1/0

y-i

V4-2/3

2/4-2t/3+!/2

3/5-33/4+32/3-2/2

3/5-43/4+63/3-4^2 + 3/1)

y\

3/5 - 2/4

V5-2J/4+2/3

Vs

The law of formation should be apparent from this table, for it
is precisely that which we meet in the binomial expansion, e.g. the
wth difference is of type

n{n—l) n{n—l){n—2)

+(-iryo.

and by equating to zero the (n-|-l)th difference we have the relation
required between the y's.

Example. — Let us apply this method to the ' Poor Relief ' example
already considered. Since there are four known points the relation
between x and y must be of the form

as before. Hence the 4th differences must vanish, and taking the

92 STATISTICS

points in order from years 1835 to 1855 as (tq, ?/(,), (:rj, y^), (^2, i/j,),
(•^^3. yi)r (^4' Va)' we get

Z/4— 4^3+%2-42/i+2/o=0
as the formula connecting five ^'s, four known and one {y.^) unknown.

Therefore %2=4(yi+2/3)— (2/0+2/4)

=4(4577+5395)- (5526+5890),

which is equivalent to equation (4) on p. 89,

Thus y„ = £4,745,000.

Third Method {by means of advancing differences). In the last
method we employed a relation connecting y^ with all the preceding
«/'s, but it is possible also to express y„ in terms of ?/o and the suc-
cessive differences, which may be written /\, A^j A^j • • • A" 5
we have, in fact, with the notation of Table (23) : —

Ao=2/i-S'o> A 0^=2/2- 2^1 +2/o> Ao''=2/3-3«/o+3ii/i-?/o, . . .

Thus

2/1=2/0+ Ao-

2/2=22/1-2/0+ Ao'=2/o+2Ao+Ao'-

2/3= %2 ~ ^^2/1 + 2/0+ A 0^

=3(2/o+2Ao+Ao')-3(2/o+Ao)+2/o+Ao'
=2/o+3Ao+3Ao'+Ao'-
2/4=42/3-62/2+42/1-2/0+ Ao^
=4(2/0+3 Ao+3Ao'+Ao')- 6(2/0+2 Ao+Ao')+4(2/o+Ao) -

2/0+ A 0'
=2/o+4Ao+6Ao'+4Ao'+Ao*-

Here again the law of formation is clear, and it is readily estab-
lished by induction that, for all 'positive integral values of n,

^ , nin—\) ^ „ , nin—\)[n—2) ^ „ , ,_.

2/n=2/o+^Ao+ \ ^ Ao'+ \ 2 3 ^° "^ ^ ^

a series which automatically comes to an end at the term Ao"-

An extension of this formula is obtained by writing 6 in place
of ?i, where 0<^<1. We then get

, . ^ 6(1-6) ^ „ , 6{l-6){2-e) ^ „ ,_,

2/g=2/o+^Ao--Y^Ao^+ \ 2 3 ^" • • •' • • • ^^^

which enables us to interpolate for a, y in between any two of a series
of y's corresponding to .r's advancing by equal steps. This relation

GRAPHS

93

is no longer identically true as was (5), for the series on the right
in (6) is unending, but its application in practice is justified when,
as the differences advance, the numbers obtained tend to grow
smaller and smaller, so that the remainder after a certain number
of terms can be treated as negligible. Unless this tendency is
reahzed without carrjing the differences far the formula is not
very satisfactory.

To illustrate the method of procedure the following figures may
be used from Table (7), p. 25: —

Table (24). Marks obtained by certain Candidates

IN an EXAillNATION

No. of

First

Second

Third

No. of Marks.

Candidates.

difference

difference

diflFerence

y

A

A2

AS

Not more than 45

447

37

„ „ 50

484

21

-16

1

»? •» )j 55

505

6

-15

12

„ ., 60

511

3

- 3

» » 5» 6o

514

Suppose now we ^dsh to know the number of candidates who
obtained a number of marks not more than 48. In that case, in
applying formula (6), we have

1/0=447, ^=(48-4o)/(50-45)=3/5,

Ao= 37, Ao'=-l<3, Ao'=l>

and hence, up to this order of differences, the required number of
candidates is given by

447+1 . 37-I^\-16)+^il^(l)
1.2^ ^1.2.3

=447+22-2+ 1-92+006

=471, approximately.

Also, number of candidates obtaining more than 48 marks, but not
more than 50

=484-471

= 13, approximately.

94 STATISTICS

Fourth Method {by means of Lagrange's Formula). We shall
consider one more formula, due to the famous French mathematician
Lagrange (173G-1813), which is useful when the recorded y's, corre-
spond to x's, which advance by unequal stages.

Let the given statistics be represented as before by

(•^0' !/o)> (-^l. yi)> (^2. 2/2). • • • K. 2/n).

and consider the equation

[X X^)\X X'2) . . . \x x^)

y=yo
-\-y

\Xq X^)\Xq X^) . . . (Xq X^)

(X Xq)[X X^) . . . (X Xf^)

(Xi Xq){Xj X2) . . . (^1 x^)

(X X^J){X Xi) . , . (X ^„-i) ,-.

+yn^ ^ ,^ , _ , — . ••.(')

\X^ Xq){X^ X^) . . . [Xj^ ^n-l)

It is of the nth degree in x, and it is identically satisfied by the
(n-|-l) pairs of values

{x^xo, y=yo), {x=Xi, y=yi), • • • {x=^Xn, y=yn)-

It will therefore clearly serve as the fitting curve
?/=Co+Ci.c+C2a;2+ . . . +c„x'»,

being exactly of this type, and in order to get the y corresponding
to any other x we have only to substitute that value of x in (7).

Example. — The following figures, based upon data from Porter's
The Progress of the Nation, show the age distribution of criminals
in the year 1842.

Percentage of criminals up to age 25=52-0 {y^).

„ 30=67-3 (1/1).
„ 40=84-1 {y.,).
„ 50=92-4(^3).

Let us employ Lagrange's formula to find the approximate
percentage of criminals up to 35 years of age, making use of the
four ordinates given, and taking a: =35. We have

_^^(35-30)(35-40)(35-50) ^^g(35-25)(35-40)(35-50)
^ "^ (25-30)(25-40)(25-50) (30-25)(30-40)(30-50)

g^^(35-25)(35-30)(35-50) g^^(35-25)(35-30)(35-40)
(40-25)(40-30)(40-50) (50-25)(50-30)(50-40)

= _10-4+50-475+42-05-4-62
=77-5.

GRAPHS

95

Number of cigarettes bought
Fig. (13).

Reasoning made Clear with the Help of Graphs or Curves. The
graphical method not only produces an instructive picture of a
scheme of observations, but it may also be used effectively on
occasion to pilot one through the intricacies of economic or similar
argument. The eye is a very ready pupil and is quick to pass on
what it sees to the mind ; it acts, that is to say, as an ally to the
understanding, which might get on A^thout it, but which certainly
gets on faster with it.

To illustrate this we shall consider the first jDrinciples of an
interesting class of curves relating y
to supply and demand.*

Curve of Demand. Conceive a
smoker who buys cigarettes at
the rate of x per day, and paj^s for
them at the rate of y pence each.
Altogether they cost him there-
fore a sum of xy pence per day,
which is conveniently measured
by the rectangle OABC in fig'. (13).
Notice that the cost price of each single cigarette is here represented
by the area {yx 1), while the total expenditure is represented by the
area {yxx).

Now let us suppose his country is at Avar and that the smoker,
to put himself in a position to discourage luxuries, decides to give

up smoking. Let us try to
measure in terms of pence the
cost of this great sacrifice to
him on the first day.

The first cigarette is probably
the hardest to do Avithout, and
the desire for it is so strong
that, if it Avere a mere matter
of monej^ and not of patriotism,
~X he Avould be Avilling to give as
manj^ pence as are represented,
say, by the rectangle 1-1 in
fig. (14) in order to haA^e it to smoke. If he went on to bargain

S-C

12 3 4

Number of cigarettes bouglit
Fig. (14).

[* A fuller account of these curves will be found in Cunynghame's Geometrical
Political Economy, where a rather more accurate interpretation of "surplus
value " is given, involving the introduction of subordinate curves. The
simplified statement here adopted seemed sufficient in an introductory course.
Marshall's Principles of Economics also contains many fascinating illustrations
of the use of such curves, mainly in footnotes. ]

96

STATISTICS

with himself in imagination, he would not be ready to offer quite
so much for the satisfaction of a second smoke soon after the
first : he would perhaps only give a number of pence represented
by the rectangle 2-2 in the figure for this second cigarette. And
if it came to a third he would offer less still, only ' 3-3 ' pence
perhaps, for the fourth ' 4-4 ' pence, and so on. The rectangles
here are of varjnng height, but each stands on a base of unit length.
Thus we find that the total sum he would be prepared to offer,
bargaining for cigarette after cigarette in this way, would be repre-
sented by the sum of the rectangles 1-1, 2-2, 3-3 ... in fig. (14),
where the addition of each unit length along OX means one more
cigarette in imagination smoked, and a diminution of unit length
in an ordinate parallel to OY means a reduction of Id. per cigarette
in the })rice the smoker would be prepared to pay.

But if he fell a prey to his persistent cra\'ing and actually bought
a number of cigarettes represented by OA in the figure, each would
cost him in the ordinary way only a number of pence represented
by AB, say, i.e. area (ABx 1), and his total expenditure Avould thus
be measured by the area of the rectangle OABC. He would get
them, that is to say, for less than he would be prepared to give
rather than go without them. The difference, the area of the
rectangles making up the portion BCDE of fig. (14), represents the
measure in pence of surplus enjoyment wliich he would obtain free

of charge, or it represents the
measure of free sacrifice he
makes if he is true to his
patriotic principles.

Let us now take an example
on a larger scale. Imagine a
small community of people,
producers and consumers, buy-
ing and selhng among them-
selves. Some of them are
coalowTiers and sell coal to
the others in the open market,
where competition is supposed free mid unrestricted in any way. This
last condition is emphasized, because it is seldom perfectly satisfied
in the real world of commerce.

Just as in the previous case we may represent the number of
cwts. of coal bought by a length OA measured along OX in fig. (15),
and the price actually paid in shillings per cwt. by the area of a
rectangle on unit base and of height OC along OY. Thus the

12 3 4 A

Number of cwts. of coal bought

FiQ. (15).

GRAPHS

97

total cost to the consumers in shillings is measured by the area of
the rectangle OABC.

But here again we may picture the consumers during a coal
shortage, when, rather than go without the first cwt. of coal, some
one among them would be ready to offer for it as many shillings as
are represented by the rectangle 1-1 in fig. (15), and for the second
c\H. some one would be ready to offer ' 2-2 ' shillings, for the third
' 3-3 ' shillings, and so on. The demand for coal could thus be
measured in shillings by the sum of the rectangles 1-1, 2-2, 3-3
. . . and, if OA runs into thousands of units of coal, the lengths
0-1, 1-2, 2-3 . . . along OX, corresponding to additions of 1 c^vt.
in the quantity bought, would in the limit be so small that the
sum of the rectangles would become practically equivalent to the
curvilinear area OAED in the figure, where DE is a curve drawn
through the summits of the rectangles, namely the curve of demand.

The consmners' surplus in this case would be measured in shillings
by the area BCDE, this being the difference between the measures
of the sum actually paid for the coal bought and the sum consumers
would have been willing to pay rather than go without it.

Curve of Supply. Now let us consider the question from the
point of view of the coalowners. We shall assume that the average
cost of production per cwt. of y
coal increases steadily as the
number of cwts. produced in-
creases ; this would not be an
unreasonable assumption in most
cases after passing a certain point, ^ «
since the richer coal measures ° :- ^
known are likely to be mined o "S
before the poorer ones, and the
cost of mining near the surface
is bound to be less than when
deep shafts have to be bored.

If, then, OA, fig. (16), represents the number of cwts. of coal
sold, and if the price in shillings per cwt. at which it is sold is de-
noted by the area of a rectangle on unit base and of height OC
along OY, the total payment received by the coalowners will be
measured in shillings by the area of the rectangle OABC.

But the cost of producing the first cwt. is perhaps measured
by the rectangle 1-1, that of producing the second cwt. by the
rectangle 2-2, the third by the rectangle 3-3, and so on, each rectangle
being draAvn on unit base representing an advance of 1 cwt. (The

G

12 3 A

Number of cwts. of coal sold
Fig. (1G).

98

STATISTICS

advance in the cost of production would not in reality be measured
by so much the cwt. of course, but the assumption is inaccurate
in degree only, not in principle, and, by making it, the argument
is rendered clearer.) Thus the actual cost of production is, in the
limit when OA is very large and divided up into relatively very
small parts, measured in shillings by the curvilinear area OAED,
where DE is a curve dra^vii through the summits of the rectangles
namely, the curve of supply.

The difference, BCDE, between the areas OABC and OAED
represents what is known as producers' surplus, for it measures the
profit made by the owners in selling the coal at a higher price than
the cost price of production.

Now let us combine the curve of supply (S.C.) and the curve of
demand (D.C.) in the same figure, fig. (17). Their meeting point

P determines the number of cwts.
of coal bought (x), and the selhng
price in shilUngs j)er cwt. (y).
For it is clear that under normal
conditions it would not be profit-
able to coal producers to pass this
point, because beyond it the de-
mand on the part of coal consumers
measured in money is less than
the cost of production : they are
not willing on the average to pay
so much as ys. per cwt. for it,
and it costs more than ys. per cwt.
on the average to produce. If,
on the other hand, the amount of coal produced decreases below
X cwts., the greater this decrease the higher does the profit become
on the sale of it, because the greater is the difference between the
cost price and the selling price ; hence, as profits become more
pronounced, recruits wiU be attracted into the coal-producing
business, and, if this goes on, deeper shafts will have to be bored
and poorer fields worked until profits begin to decrease again and
the supply once more approaches x cwts. Thus sooner or later
the j)roduction of coal and its market price will tend to the level
determined by the equilibrium point P where the supply and
demand curves meet.

Endless varieties of problems may be discussed by altering the
conditions and observing the effect produced in the standard
diagram. Three examples Mill suffice to illustrate the method.

Number of cwts. of coal bought or sold

S.C = Stipplij curve

D.C. -Demand curve

Fig. (17).

GRAPHS

99

1. Effect of a Change in Normal Demand. Here we suppose the
normal conditions of supply are unaltered — it costs just as much
as before to produce the same amount of the commodity in question ;
but a more eager demand on the part of consumers shows itself in a
readiness to purchase more at any given price than would have
been purchased under the old conditions : this may conceivably
be due to a general increase in the purchasing power of these con-
sumers, or it may be the result of a shortage of some other com-
modity which causes this one to be more widely used, just as
margarine, for instance, has been known to take the place of butter ;
whatever the reason may be, the effect is that the demand curve
now occupies a higher level throughout its length, D'C in place of
D.C. in the figures.

When we turn to the supply side of the question, there are three
Y

Decreasing Return

stages which, although they shade into one another in practice, it
is well to separate clearly in theory : (1) the only supplies immedi-
ately available are those actually in the hands of dealers ; (2) to
meet the increased demand, and so earn for themselves increased
profits, manufacturers will speed up production, b}' working over-
time, etc., with the help possibly of any disengaged labour or
capital they may be able to secure, and the resulting extra supplies
will be available after a short time ; (3) if the demand continues
unabated manufacturers, by offering higher wages and interest,
will seek to attract fresh labour and capital from other engagements
into their business, and, by renewing their machinery and generally
improving their organization, they will produce on a larger and
relatively more economical scale. Moreover, other manufacturers,
seeing the profits to be earned. wiD be attracted into the same line
of business also, so that by this time the current available supplies
of the commodity may exceed very appreciably their old figure.

100

STATISTICS

But all this happens only in the long run, and the economist has
always to bear this extremely important element of time carefuUy
in mind when he seeks to estimate the effects of any proposed
action.

We assume then that the new demand remains long enough at
its higher level to allow for the gradual adjustment in this way of

supply to the changed
conditions, and for the
economic forces called into
play once again to arrive
at a balance between
them, most likely at a
new equilibrium point.
The two figures illustrate
the difference in effect
according as the produc-
tion of the commodity is
ncreasing e urn subject to a decreasing or

an increasing return, i.e. according as the cost of production rises
or falls when the amount produced is increased. In both cases it
will be noted that more of the commodity is produced (ON' in place
of ON) in answer to the keener demand, but the difference is much
greater in the second case than in the first. Also the jorice has
gone up in the first case, while in the second it has gone down,
the difference being measured by the change in PN.

2. EJfect of a Tax. If the Y
tax is at the rate of so much
per unit (say Is. per unit, if
the price is measured in shil-
lings) of the commodity pro-
duced, this will raise the
supply curve, S.C., bodily up
a distance of 1 unit into the
position S'.C, fig. (18), be-
cause the effect is the same
as if Is. were added to the
cost of each unit in produc-
tion. The production will
thus be diminished by N'N units, for P' is the new equihbrium
point ; the selling price will be increased by P'Ms per unit — by
less, it should be noted, than P'Q or K'K, the amount of the tax ;
producers' surplus, which is analogous to what economists term

GRAPHS

101

-en/, is diminished by (area KPL— area K'P'L')s ; consumers'
surplus is diminished by (area PLL'P')s ; finally, the tax produces
for the Treasury a number of shillings represented by a rectangle
with sides of length ON' and KK'.

3. Effect of a Monopoly. A monopolist has the power to stop
production short of the true equilibrium point, so that ON' cwts.,
fig. (19), are produced in place of the ON cwts. which free competi-
tion would demand. The selling price is thus raised by Q'Ss. per
cwt. ; producers' surplus is increased by (area KP'Q'M'— area
KPL)s ; while consumers' surplus Y
is diminished by (area PLD— area
DM'Q')s.

A word of explanation is neces-
sary before leaving the subject of
these supply and demand curves.
It is probable that the reader will
have questioned the possibility of
draAving such curves for any com-
modity with sufficient accuracy to
be of any value, but it Avould be
enough as a rule to be able to estimate what would happen
if a slight variation occurred in price or in production, and such
an estimate may sometimes be made by actual trial : e.g. a good
practical farmer most Hkely knows nothing about supply and
demand curves as such, yet from past experience he has a pretty
shrewd notion as to how far it may be profitable to spend an extra
pound here in rearing calves and a pound less there in cultivating
crops, bearing in mind the prices which cattle and com might be
expected to fetch. From his point of view the interest of the
curves, if he knew anything of them, would be centred in those
portions which correspond to normal conditions, i.e. somewhere in
the neighbourhood of the equilibrium point under the free play of
ordinary competition. '

Their real value, however, as suggested at the beginning, does
not consist in the practical assistance which they afford to the pro-
ducer or consumer, by way of foretelling the actual measure of
consumption or production, so much as in the fight they throw
upon general tendencies which are rather apt to be obscured if they
are ponderously presented with elaborate economic argument.
They make plain in a moment to the eye what can only be stated
in two or three pages of writing.

CHAPTER X

CORRELATION

One of the most important questions which can be discussed by
statistical methods is that of possible connection, or correlation, as
it is called, between two sets of phenomena. If some factor in
each can be isolated and measured numerically, our object is to
discover if the size of either is sympathetically affected when a
change occurs in the size of the other ; or, to put the matter in
another way, do large values of the one factor go with large values
of the other factor and small with small, or vice versa ? And, if
some mutual dependence of this kind exists, can an estimate of
its extent be made ?

Consider, for example, the factor or character of height in husband
and wife. Is there any connection between stature of husband (x)
and stature of wife (y) ? Do tall men tend on the average to wed
tall women, or do we find tall men choosing short women for wives
just about as often as they choose tall women ? When correla-
tion exists we shall want some measure for it which will tell us

the amount of change or devia-
tion from the average in either
character associated with a given
change or deviation from the
average in the other.

In studying graphs we saw how
some hint of the existence of
correlation might be discovered,
but we wish now to go a little
more deeply into the subject.
The first step is to measure an
adequate number of pairs of values, x and y, of the characters
concerned in order to find what values are associated together,
and how frequently the same values are repeated. When this is
done we can draw up a table of double entry, see fig. (20), setting
out in rows and columns the frequencies observed. An examina-
tion of Table (25), showing the variation of brain weight with age

102

-Vi

.V,

^3

^/

^1

-j;

^'f

Fig. (20).

CORRELATION

103

in the case of 197 Bohemian Avomen, Mill make clear what is meant.
The x's from x^ to a;^, and the i/s from y^ to y,^ are supposed to
ascend in magnitude, and when, for example, the pair of values
(^2> ^3) ^s observed to be repeated nine times, the number 9 is placed
in the second column and third row of the table, so that the frequency
of each class is found recorded in the square proper to it : thus,
out of the sample in Table (25), there are 10 women between the
ages of 40 and 50 with brains weighing between 1300 and 1400
grams.

Table (25). Variation of Brain Weight with Age in the
Case of certain Bohemian Women.

[Data from Biometrika, vol. iv. pp. 13 e^ seq.. Variation and Correlation
in Brain Weight, by Raymond Pearl.]

Age in years

^1
20-30

^2
30-40

^3
40-50

50-60

60-70

70-80

Totals

CO

J3
Si

-s:
5

.i
'5

y.
1000-1100

1

-

1

1

-

-

3

1100-1200

2

2

4

2

5

4

19

^3
1200-1300

28

9

8

14

10

4

73

1300-1400

26

14

10

6

5

4

65

1400-1500

13

7

7

2

-

2

31

1500-1600

2

3

-

1

-

-

6

Totals

72

35

30

26

20

14

197

Mean y

1325

1350

1310

1285

1250

1279

When each class interval, as in this table, includes a small range
of' values, the x and y may, as an approximation, be taken as the
mid values of their class intervals : y^ would be taken, for instance,
as 1250, though it really includes all values between 1200 and

104 STATISTICS

1300 grams. Strictly in such cases each suigle observation is not,
geometrically speaking, located at a definite point, but lies some-
where within a small area, though it is treated as if it had the values
X and y which apply to the centre point of the area. It is some-
times possible to correct for this assumption by what is loiown as
Sheppard's adjustment, but we shall not concern ourselves with
the correction in the present discussion, so as to avoid complications,
because the difference made is not generally large.

The table, when drawn up, may immediately suggest some
intimate connection between x and y. It may indicate that as
X increases y also in general increases, or that y tends to fall in
value as x grows bigger. But a more refined analysis is necessary.
It would be instructive perhaps to travel along the row of x's, find-
ing what mean value of y is associated with x^, what mean value
of y is associated with x^, and so on. This would give a sounder
basis for judging whether, as x increased, y in general increased or
decreased as the case might be : for example, in Table (25) the
m.ean values of y associated \\dth the several types of x are shown
in their proj^er columns at the foot of the table and clearly, as
X increases, y tends to decrease, apart from conflicting readings at
the beginning and end of the table, and the latter of these may not
be significant of any real difference in brain weight at the end of
life, for it is only based on fourteen observations ; generally, the
inference from this table would be that the weight of the brain
decreases as the age increases after maturity is once reached,
although, of course, it would be rash to make more than a tentative
statement with so small a sample at our disposal.

Let us suppose ?/i to be the mean value of y associated with x^,
y^ the mean value of y associated with x.^, y^ with x^, and so on.
If these values [x^, y^), (.To, y-z}, {^z^ Vs)' ^^'^■' ^^^ plotted, it is very
often found that they cluster more or less closely about a straight
line, see fig. (21), so that we are led to ask whether there is not
some line which will very fairly describe the run of the points ;
the equation of such a line would be

y=mx-\-c,

and if m and c were known we could find from this equation the best
average value of y corresjoonding to any given x.

But, on reflection, ^u ^2' Vz • • - '""^^^ themselves only the best
?/'s corresjoonding to the particular values x^, x^, x^ . . . oi x, so
that the problem is really the same as that of finding the relation

y=mx-\-c,

CORRELATION

105

based on all the observations, which A\'ill enable us to estimate the
best y corresponding to any given x.

Now for any vahie x^ of x the value of y given by this relation
is (maJi+c), while by observation we may find more than one value
of y corresponding to the value x^ of x. If y^ be one such value
the difference between it and the value given by the above rela-
tion is

(mxi+c)— t/j.

This difference we may regard as the error made in estimating y
from the relation instead of taking the value given by observation

Y r> ' li, ' u

IJj;li,v'U

witig-theJ^eanBt-ein-|Vv^ig.^.o| ,

> . asso :ia ;e

d wlh vaiin,^ Ajiit

ypes

!s ^

1

s.

S

V

CO '32^ * *S^

R 5

V :

c •

N

5

^k

\

&>

^

^S. -

§ '275

^.

^

s:

V

e

^V

^y.

s

\

^.

>^ _

s, .

^w

>,

1220

O20

30

40

50 60

Age in years

Fro. (21).

70

80

which for the moment we think of as the true value. The best
relation will then clearly be the one wliich makes all such errors of
estimate as small as possible. But, algebraically, some of these
errors are positive, i.e. the value of y given by the relation is greater
than that given by observation, and some are negative, and it is
only their magnitudes that we wish to take into account. Accord-
ingly we follow the method used in finding the standard deviation
in order to get rid of the ambiguities of sign : we form, that is to
say, the sum of the squares of the errors, because the expression so
formed will clearly be least when each separate error is as small as
possible in absolute magnitude.

106 STATISTICS

To find, then, the values of m and c which will make

{mx^+c—y^f+{mx^_+c—y^Y-\- . . . +(m.r„+c— ?/„)2

a minimum (see Note 7 in th3 Appendix), where n is the total
number of pairs of observations.

The required values are given by differentiating, first with regard
to c treating m as constant, and then with regard to m treating c
as constant, putting each result equal to zero. Thus

(ma;i+c-?/i).ri+ . . . +(m.T„+c-v/„).x-„=0

Therefore m{Xi+ . . . +.r„)+wc— (?/i+ . . . +yn)=0

w(.V4- . . . -|-,r„2)4-c(.Ti+ . . . +xJ-{x^iji-\- . . . x„y„)=0.

The first of these equations gives

'm{nx)-\-7ic—{ny)=0,
i.e. J>.- mx-\-c—y=0,

where x is the mean of all the x's and y is the mean of all the y's,
and it expresses the fact that the line y=mx-\-c passes through
the point {x, y).

This might have been expected, for, graphically, each pair of
observations (a^i, ?/]), (a;2, 2/2)' (^3' 2/3) • • • corresponds to some point,
and if we look for the line y=^mx-\-c passing through the region
where they cluster most thickly together we should certainly expect
it to pass through their mean or centre of gravity {x, y). This
suggests how the values of m and c may be considerably simplified.
If we measure aU the x's from x, their mean, and all the ?/'s from y,
their mean, which is equivalent to taking the point {x, y) as origin
and replacing every x by its deviation f from x and every y by
its deviation '*} from y, the first of the above relations is reduced
to c=0. and therefore the second becomes

Hence

where p is the mean of all the product pairs |>/, and 0-3. is the standard
deviation of all the x's.

m(^,2+ . . .

+^«')-(^i'/i+ • •

• +^„VJ=0.

w»=(^l^l+ .

. . +LVn)l{L'+ •

• . +in')

=nplnax^

=PI(Tx^,

CORRELATION 107

Thus the required equation for estimating the best >? correspond-
ing to any particular f is

whence y—y= — {x—x) . . • (1)

The coefficient pja^^ in this equation evidently gives the deviation
in y from the mean y corresponding to unit deviation in x from
the mean x, for when (a;— x) = l, {y—y)=p!(J^. Hence the greater
this coefficient is, the greater will be the change in y resulting from,
or at all events coexistent with, unit change in x.

Thus Jpjcrj' would seem to supply a not uiu'easonable measure of
the correlation between x and y. But there is something very
unsymmetrical about this result. Why should the correlation be
measured by pjcT,^ any more than by l^loy^ ? In fact, we might
repeat the whole of the previous argument, interchanging x and y
throughout wherever they appear. In that case we should first
travel down the column of y's and calculate the mean values of x
associated with yi- yi-, y^ . . . respectively. This would give a set
of points {xj^, y^), {x^^, y^), {x^, y^), . • . , which, when plotted, would
perhaps lie approximately in a straight line. We should thus be
led to look for some relation

x=m'y-\-c'

which would enable us to estimate the best average x corresponding
to a, y oi given type, and, proceeding just as before, we should
ultimately obtain the equation

or ^x-x)=l{y-y), . , . (2)

in which the coefficient J)l(^y~ gives now the deviation in x from the
mean x corresponding to unit deviation in y from the mean y.

Hence pfay^ has, seemingly, Just as much claim as j:)/^^;^ to measure
the correlation between a* and y. The one gives the change in x
corresponding to unit change in y : the other gives the change in y
corresponding to unit change in x ; and the only reason why they
differ is because unit change in x does not mean the same thing as
unit change in y : their standards of changeableness or variability
are not equal. If then we could alter the scales of measurement
so that unit change in each were of the same magnitude, the two
coefficients obtained ought to become identical, and we should then
have a really satisfactory measure for the correlation required.

108 STATISTICS

With this object let us examine the variabiUty of the x's and
compare it with the variability of the y's. Now the total dispersion
of the different x's on either side of x, the mean x, is conveniently
measured by a^., their standard deviation. And similarly the
dispersion of the y's on either side of y, the mean y, is measured
by cTy. The bigger CTj. is, the greater is the variability of the x's,
and the bigger ay is, the greater is the variability of the y's. Hence,
in equations (1) and (2), (x—x) should be divided by o-a; ^^^^ iy~y)
by ffy if we want to work with the same unit of change or variability
in each case. The equations then become

y-

'—y\_ p /x—x\

^y I ^x^y^. ^x I

x—x\ p iy—y
and ' \ -f IJ J

Write r=p/(Tjfry ; then r is taken to be the coefficient of correla-
tion, for it measures the change in either character corresponding to
unit change in the other when the tmits are made comparable.

The lines giving the best y for a given x and the best x for a
given y may now be "wnritten

y—y^r-'ix—x)

and x—x~r-{y—y),

Oy

and they are called lines of regression. The term regression was
first used by Sir Francis Galton in a paper entitled Regression
towards Mediocrity in Hereditary Stature, though the root idea
is not by any means confined to characters affected by heredity :
it holds for any pair of correlated variables. Galton found that
if a number of tall fathers are selected and their heights measured*
the mean height being calculated* and if, further, the heights of the
sons of these fathers are measured, their mean height being like-
wise calculated, the latter is not equal to the mean height of the
selected fathers, but is rather nearer the mean height of the popula-
tion as a whole. There is, that is to say, a regression or stepping
back of the variable towards the general average. Professor Karl
Pearson has remarked that ' in the existing state of our knowledge
the recognition that the true method of approaching the problem
of heredity is from the statistical side, and that the most we can
hope at present to do is to give the probable character of the offspring

CORRELATION 109

of a given ancestry, is one of the great services of Francis Galton to
Biometry.'

The expressions r— and r— are called coefficients of regression,

and they register in the above particular case the amount of abnor-
mality to be expected in the height of the sons when the amount of
abnormality in the height of the fathers is known, and vice versa.
The regression of the sons' height, y, on the fathers' height, x, is, ^^j^i
in fact, defined as the ratio of the average deviation of the heights """^^
of the sons from the mean height of all sons to the deviation of the 5

heights of the fathers from the mean height of all fathers, and hence
it may be written y __

To make the definition more general, instead of speaking merely in
terms of height, we refer to any row or column— for there is no
intrinsic difference between row and column — in a table like
Table (25) as an array of ?/'s or of .f's, and selecting a particular
type, say a particular value of x (like fathers of height x), we define
the regression of the corresponding array of y's (like heights of sons
of these fathers) on the type x to be the ratio of the average devia-
tion of the array of y's from the mean y to the deviation of the
selected type x from the mean x.

Example. To illustrate, let us take some figures due to Professor
Pearson and Dr. Alice Lee [Biometrika, vol. ii. pp. 357 et seq., On
the Laws of Inheritance in Man]. Suppose the mean stature of all
observed fathers, based on a sample of over 1000 observations
= 67-68 in., with S.D.=2-70 in.

Also suppose the mean stature of all sons= 68-65 in., with S.D.
--2-71 in., and that the correlation r between stature of father
and stature of son =0-5 14.

The regression of son on father as regards stature is then given by

(;?/-68-65)=(0-514)^— (.r-67-68)

where x is the height of selected fathers and y the mean height of

their sons.

Hence ?/-:0-516x+33-73,

so that if we selected fathers of height 70 in., for example, the
mean height of their sons would not be 70 in., but

(0-516)(70)+33-73=69-S5 in.,

110 STATISTICS

i.e. there is a regression towards the general mean, 68-G5 in., of
all sons.

Also the coefficient of regression

= (0-514)(2-71)/(2-70)

=0-516.

It is not difficult to show that the greatest numerical value r
can in general take is unity, for consider the expression for the
sura of the squares of the differences between the observed devia-
tions of the y characters from their mean and the corresponding
deviations as deduced from the best fitting regression line,

y—y=r^{x—x).

If, with our previous notation, '/ denote the observed deviation of
the one character y, associated with a particular deviation, f, of
the other character, x, then, since {rayja^)^ denotes the best value
given by the line, the sum of the squares of the differences between
these values

= (V+ • • • +'/«'')-2r- (^x^i+ . . . +|„VJ+r^— ,(1,2+ . . . +|„2)

2

=nay^—2r-^{nra^ay)-^r^^{na^^)

= ?icr/(l — r^).

Since the sum of a number of squared quantities must be positive,
it follows that r^ must be less than I and hence r lies between — 1
and -fl.

Further, n<jy^{l — r^) can only vanish if every one of the squared
quantities on the other side vanishes independently of the rest,
so that we only get r=±l, when

'}ilii=''hlL= ' - ■ =''1nlL=ray!a^.

In this case the deviation of the one character from its mean is
always exactly proportional to the deviation of the other character
from its mean, and the correlation is then said to be perfect, for
it is equivalent to causation. In perfect correlation a one-to-one
correspondence thus exists between the values of the two char-
acters, for to one value of either there corresponds one and only
one value of the other and the standard deviation of the array

CORRELATION

111

(measuring its variability) corresponding to any selected type
vanishes.

Zero correlation is at the opposite extreme where, no matter
what the type selected in the one character may be, the mean
value of the array in the second character is unaffected, because
the two characters are quite independent or uncorrelated ; the
deviation of y from its mean bears no relation at all to the deviation
of X from its mean, and unit change in either is associated with no
particular change in the other, so that r must in this case be zero.

When r is negative, since {y—y)j{x—x) = rayj(jx and the ct's are
necessarily positive, corresponding to any value of x above the
mean of all the a;'s the best value of {y—y) is negative, that is, the
best value of y is below the mean of all the y's, and vice versa.
This means that in general high values of x would be associated
with low values of y, and vice versa.

If we take the mean as origin so that the regression lines become

y=rayla^ . x,
x=rajay . y,

these lines coincide with the axes when the correlation is zero,
and with one another when r=±l and the correlation is perfect,
fig. (22). Given two equally
variable characters [a^^^ay) and
perfect correlation, the regres-
sion lines coincide with one of
the bisectors of the angle formed
by the axes.

It may be helpful to look back
again now at the graphical view
of the argument leading up to
the determination of the co-
efficient of correlation. For

successive values of x we calculated the means of the several
2/'s observed, these being presumably the best available ?/'s corre-
sponding to the particular a;'s selected, and we assumed that,
when plotted, the points so obtained, {x^, y^), {x2, y^), (^3, 2/3), • • •,
lay roughlj' in a straight line. In the same way we calculated the
means of the several a:'s observ^ed to correspond to particular y's,
selected, and again we assumed that the resulting points, {x-^^, y^),
{xo, y-i), (^3, 2/3) •• • lay roughly in a straight line. These assump-
tions are justified in very many cases, but when they fail recourse
must be had to other methods beyond the scope of this book. [See

(Mean) X

degression Lines when
Correlation is perfect (r=+i)
Fio. (22).

112

STATISTICS

for example, Pearson's paper in Drajpers' Company Research Memoirs
Bionietric Series ii., On the Theory of Skew Correlation atid Non-
linear Regression, introducing the correlation ratio, v. which is
equal to r in the particular case when the regression is Unear.]
Sometimes, again, although the observations are so scattered that
the assumption of a straight line to describe the best fit seems
somewhat wide of the mark, it may be justified on the ground that
no better graphical result would be given by using any other curve
in place of the line. Moreover the linear expression, y=--mx-\-c,
is simple and may serve to give at all events the first two terms of
some more complex relation supplying an estimate for the most
probable y corresponding to a given x.

If we had plotted all the original pairs of observations, instead
of plotting certain x's and the mean y's associated with them, or

certain y's and the associated mean
x's, the two lines of regression would
not have stood out so clearly : they
would have lacked definition, like an
optical image which is not strictly in
focus, but there would have been a
concentration of observations, as of
light, in the neighbourhood where the
lines of regression intersect, namely
at {x, y), the mean of all the x's and
all the y's. When, however, the lines of regression lie close together
they become more clearly defined, all the observations being centred
then more nearly in one line, and the correlation tends towards
perfection. Such cases are frequent in Physics but rare, if found at
all, in that class of Statistics into which the element of human
impulse enters. When r is less than 1 the lines of regi'ession, if the
regression is of linear type, will be inclined to one another at some
angle between 0 and 90 degrees.

If only a rough value of r, the correlation coefficient, is required,
that may be obtained by merely estimating the gradient of each
regression line and multiplying the results together, one measured
relative to the axis of x and the other relative to the axis of y,
for this product

= (regression of y on x) (regression of x on y)

/a,

= r-

CORRELATION

ii;

Such an estimate may also be useful, though it may not be very
dependable, when the complete distribution of characters is not
known, for either regression line can be drawn when any two points
on it are known and a single array of values of either character
corresponding to a given type of the other is sufficient to fix one
such point ; also the mean {x, y), if it were known, would at once
give a j)oint common to both regression lines. When all the facts
are available, however, the method of calculation is to be preferred
to that of simply graphing tlie observations and their means, as there
is bound to be a certain amount of guesswork and consequent error
in deciding from a graph how the best regression lines run.

It is frequently convenient to refer the deviations of the given
variables to some jDoint other than the mean {x,y) as origin, and,
when this is done, a correction
must be applied to the resulting
value of r. We have already
explained how, in such a case,
to correct for standard devia-
tions, and, as r^ployPy, it only
remains to explain how to cor-
rect for ji-

Now i^t is given by

where the |'s and '/'s denote deviations from x and y respectively.
Fig. (23) indicates the changes necessary in transferrmg from some
origin 0 to the mean G. The co-ordinates of P (representing a
typical observation) referred to 0 are {x, y) and referred to G are
(^, ■,]). Also the point G itself referred to 0 is {x, y). Thus

^=-x—x, v-^y-y,
and np becomes

{xi-x){ij^-y)+ . . . -^{x„'-.f)iyn-y)

= {x^yi-xy^-yXi^xy)-\- . . . +(a;„y„-%„-j^a:n+a:^)

= (:c3t/i+ . . . -\-x^y,,)-x{y^-\- . . . +y„)-^(.^i+ . • . +.Tj + na-j/

= (Xi2/i+ . . . +x„yj-x.ny-y .nx-\-nxy

=Z{xy)—nxy,

where S{xy) denotes the sum of expressions of the type xy.
Hence the corrected value of 2?

^E{xy)jn-xy,
H

Y

y'

G

J'

X

•

P

^ : X-

y

O

^

-X >-

X

Fig. (23)

114 STATISTICS

from which we infer that the corrected value of r is

E{xy)—nxy

We proceed to a few appUcations of these results in the next
chajDter.

[As early as 1846 a French physicist, Auguste Bravais, had conceived the
surface of error as a means of describing in space the path of a point whose x
and y co-ordinates are subject to errors which are not independent ; but it
appears to be doubtful whether he saw the connection between his work and
the subject of correlation. It was Galton, nearly forty years later, who
really created that subject, introducing the coefficient of correlation on graph-
ical lines and giving practical examples of its use. (See Biometrika, vol. xiii.,
pp. 25-45, Notes on the History of Correlation.)

Edgeworth, in 1892, using Galton's function, independently reached some
of Bravais' results related to the correlation of three variables, and showed
how they could be extended. Karl Pearson, in 1896, contributed to the
Royal Society Transactions a fundamental paper on the subject, with special
reference to the problem of heredity, drawdng attention to the best value of
the correlation coefficient, and how it should be calculated. (See Appendix,
Note 11.) Yule, returning in the following year to Bravais' formulae, showed
their significance also in the case of skew correlation.

Pearson afterwards developed a method of determining the correlation of
characters not quantitatively measurable, and in a discussion of the general
theory of skew correlation in another paper he proposed a new function, the
correlation ratio, applicable to the case of non-linear regression.]

CHAPTER XI

CORRELATION EXAMPLES

Example (1). — To find the correlation between Differences in Whole-
sale Price Index Numbers and in the Marriage Rate from their corre-
sponding Nine-yearly Averages during the twenty years, 1889-1908.
using tlie data given on p. 77.

Table (26). Correlation between Differences in Wholesale
Prices and Marriage Rate from their respective Nine-
yearly Averages.

(1)

(2)

(3)

(4)

(5)

(6)

Year.

Difference in
Prices from

Square of
No. in

Difference in

Marriage-rate

from 9-yearly

Average.

Square of
No. in

Product of Nos.
in Col. (2) and

9-yearly Average.

Col. (2).

Col. (4).

Col. (4).

{X)

(x2)

(Z/)

ir)

(^.y)

1889

+ 0-9

0-81

+ 1

1

+ 0-9

1890

+ 2-3

5-29

+ 6

36

+ 13-8

1891

+ 7-0

49-00 '

+ 6

36

+ 42-0

1892

+ 2-4

5-76 !

+ 3

9

+ 7-2

1893

+ 2-0

4-00

- 6

36

-12-0

1894

- 2-8

7-84

- 5

25

+ 14-0

1895

- 4-3

18-49

- 6

36

+ 25-8

1896

- 6-1

37-21

+ 1

1

- 61

1897

- 3-7

13-69

+ 3

9

-11-1

1898

- 0-2

004

+ 4

16

- 0-8

1899

- 1-6

2-56

+ 6

36

- 9-6

1900

+ 5-3

28-09

+ 1

1

+ 5-3

1901

+ 1-0

1-00

1902

- 0-5

0-25

+ i

1

- 0-5

1903

- 1-4

1-96

- 1

1

+ 1-4

1904

- 1-3

1-69

- 3

9

+ 3-9

1905

- 2-4

5-76

- 2

4

1 + 4-8

1906

- 0-5

0-25

+ 3

9

- 1-5

1907

-f 3-2

10-24

+ 6

36

+ 19-2

1908

- 1-8

3-24
197-17

- 2

4

+ 3-6

-f 241 -26-6

+41-25

306

+ 141-9-41-6

116 STATISTICS

The arithmetic is comparatively simple in this case because
there is only one value of each variable corresponcUng to each year,
so that there is no weighting or grouping to complicate the analysis.
The variables x and y, between which we wish to find the correlation,
appear in col. (2) and col. (4) in Table (26), and the positive and
negative differences are sej)arated from one another in each case
so as to make their summation easier.

Thus for the arithmetic mean of the numbers in col. (2), wc have

.r=(+24-l-26-6)/20=-0-125 ;
and for the mean of the numbers in col. (4), we have

^=(+41-25)/20=+0-8.

The straightforward procedure would now be to get the twenty
corresponding values of ^ and V, the deviations of the twenty x's
in col. (2) and of the twenty y's in col. (4) from x and y respectively,
and, having found o-j, and Gy, we could immediately deduce r from
the formula

= (fi^i+ . . . +f2o^/2o)/20a^a,„.
But it is simpler to measure the deviations from (0, 0) as origin
rather than from the mean (—0-125, +0-8), because x^, y^, and xy
involve fewer significant figures than would ^^. v'^, and f v, and,
of course, it will be necessary to correct for this at the end in the
usual way.

The mean square deviation of x referred to zero as origm
= 197-17/20, by col. (3).

Therefore, cr^^^ 197-17/20- (0-125)2=9-843

a, =3-14.

Again, the mean square deviation of y referred to zero as origin
= 306/20, by col. (5).
Therefore, ct/= 306/20- (0-8)2= 14.66

(7^=3-83.
Also the coirected p

= {Exy)ln—xy

= 100-3/20- (-0-125)(+0-8), by col. (6)

= 5015+0-100

= 5-115.

Hence f—Vl^^x'^v

= 5-115/(3-14)(3-83)
=0-43.

CORRELATION — EXAMPLES 117

It is necessary to be careful with the signs in forming the numbers
in col. (6), but otherwise the actual calculation should present no
difficulty.

The regression equation giving the best marriage rate difference,
Y, for a given wholesale price difference, X, from their respective
nine -yearly averages is

(Y-0-8)=r^^ . (X+0-125)

= (0-43)^--— \x+0-125)
(3-14)

i.e. Y=0-52X+0-86.

The regression equation giving the best wholesale price difference,
X, for a given marriage rate difference, Y, from their respective
nine-yearly averages is

(X+0-125)=r^ . (Y-0-8)

=0-35(Y-0-8)
i.e. X=0-35y-0-40.

We noted that fig. (10), p. 80, suggested a closer correlation
between the two factors we have been considering during the
earlier years of the period 1875-1908 than during the later years.
It might be worth while as an exercise to see if this is borne out
by calculating r for the years 1875-1889, and comparing it with
the value found for the years 1889-1908.

Example (2). — To find the correlation between Overcrowding and
Infant Mortality in Lotidon Districts. [Data taken from London
Statistics, vol. 23, published by the London County Council.]

The figures are apparently based ui^on the Census Report of
1911. The numbers in col. (2), Table (27), show what percentage of
the total population occupying private houses in each district were
living in overcrowded conditions, any ordinary tenement which
has more than two occupants to a room, including bedrooms and
sitting-rooms, being defined as overcrowded. The numbers in
col. (5) show the infantile mortality in each district, that is, the
number of infants who died under one year out of every 1000
bom, including both sexes.

For the sake of comparison these numbers have been plotted
together on the same graph sheet. The districts, arranged in
alphabetical order, were numbered from 1 to 29 so as to form a hori-
zontal scale corresponding to the scale of years in discussing prices
and marriages. The scale in this case is, of course, purely artificial,

118

STATISTICS

and the only reason for joining up neighbouring points is that we are
better able by so doing to see whether or not high values of the one
variable go ■with high values of the other variable, and low with low.
In calculating the mean and standard deviation for overcrowding
we have measured deviations from 17-0 as origin, and in making the
same calculations for infant mortahtj' we have measured dela-
tions from 125 as origin. It is convenient, therefore, to use the
point (17-0, 125) as origin in working out also the product deviation
sum, col. (8) of Table (27), instead of using the mean (17-86, 126).

Table (27). Correlation between Overcrowding and
Infant Mortality in London Districts (1911).

(1) (2) (3) (4) (5) (6) (7) (8)

Per-

centage
of

I 1
Deviation of j Square

Infant ; De\iation of

Square
of No.

Product of Nos.

District.

Popula-

No

in Co\. (2) ' of No. in

Mor-

No.

in Co

.(5)

in Col. (3) and

tion
Orer-

from 17 '0.

CoL (3).

taUty.

from 125.

in
Co1.(6)l

Col. (6).

crowded

M

(«/)

(1) Battersea .

13-3

- 3-7 1-3-69

124

1

1

+

3-7

(2) Bermondsev

23-4

-i-

6-4 40-96

156 ; +

31

961

+

19S-4

(3) Bethnal Green .

33-2

-1-

16-2 ' 262-44

151 +

26

676

-f

421-2

(4) Camberwell .

• 13-5

- 3-5 12-2.5

109

-

16

2.56

+

56-0

(5) Chelsea .

14-9

- 2-1 4-41

109

-

16

256

'

33-tJ

(6) City of London

12-3

- 4-7 22-09

124

_

1

1
289'

+

4-7

(7) Deptford .

12-2

- 4-8 23-04

142

17

- 81-6

(8) Finsburv .

39-8

+

22-8 519-84

1.56

31

961

-I.

706-8

(9) Fulham" .

14-6

- 2-4 5-76

125

(10) Greenwich

121

- 4-9 24-01

128

-L.

3'

"9

"- 14-7

(11) Hacknev .

12-4

- 4-6 21-16

119

_

6

36

27-6

(12) Hammersmith .

14-2

- 2-8 7-84

146

+

21

441

- 58-8

(13) Hampstead

71

- 9-9 98-01

78

-

47

2209

+

465-3

(14)Holborn .

25-6

+

8-6 73-96

115

-

10

100

- 86-0

(15) Islington .

20-0

+

3-0 1 9-00

127

+

2

4

-f

6-0

(16) Kensington

17-1

+

0-1 0-01

133

+

8

&4

-f

0-8

(17) Lambeth .

13-6

- .3-4 11-56

123

-

2

4

-L.

68

(18) Lewisham.

3-9

- 1.3-1 171-61

1 104 '

-

21

441

J-

275-1

(19) Paddington

16-2

- 0-8 0-64

127 -f

o

4

- 1-6

(20) Poplar .

20-6

+

.3-6 12-96

157

-f

32

1024

-1-

115-2

(21) St. Marylebone

20-7

+

3-7 i 13-69

108

-

17

289:

- 620

(22) St. Pancras .

25-5

+

8-5 ' 72-25

112

—

13

169

-110-5

(23) Shoreditch

30-6

-1-

19-6 384-16

170

-r

45

202;5

-f

882-0

(24) South wark

25-8

O-

8-8 77-44

144

+

19

361

-T

167-2

(25) Stepney

35-0

+

18-0 324-00

144

+

19

361

+

3420

(261 Stoke Newington

8-8

- 8-2 67-24

102

_

23

529

+

188-6

(27) Wandsworth .

6-3

-10-7; 114-49

122

—

3

9

-f-

321

(28) Westminster .

12-9

- 4-11 16-81

103

—

22

484,

+

90-2

(29) Woolwich.

6-3

-10-7| 114-49

'<

97

""

28

784

+

299-6

+ 119-3-94-4

2519-81

i

+

2.56-

226

12748

+ 4322-9-416-1

CORRELATION — EXAMPLES

119

For overcrowding,

mcan= 17+24-9/29= 17-86 ;

c^x= V[ (2519-81/29)- (0-86)2]= v'(86-15)=9-3.

For infant mortality,

mean=125+30/29= 126-03;

a„==V[(12748/29)-(l-03)2]=V438-5=20-9.

Ahop, referred to (17-0, 125)=(4322-9-416-l)/29-3907/29, and,
referred to the mean (17*86, 126-03), this becomes

= 3907/29- (0-86)(l-03)

= 133-8.
Hence /•=133-8/(9-3)(20-9)=0-69.

so that the correlation between overcrowding and infant mortality
is fairly marked.

1 12 13 14 15 16 17 13 19 20 21 22 23 24 25 26 27 2S 29

Numbers representing various London Districts
Fu>. (24).

The regression equation giving the average infant mortality, Y,
for districts in which the extent of overcrowding, X, is known is

¥-126-03=

:r'-'(X- 17-86)

Ox

(0-69)(20-9)

(X- 17-86)

9-3
i.e. Y=l-55X+98-4.

Similarly, the regression equation giving the average j)ercentage
of overcrowding, X, for districts with a known amount of infant
mortahty, Y, is

X- 17-86=r'^^(Y- 126-03)

t.e.

= 0-31(Y-126-03)
X=0-3iy-21-0.

120

STATISTICS

Example (3). — The reader might apply the same method to the
determination of the correlation between Ratio of Indoor Paupers
and Ratio of Outdoor Paupers, each measured per 1000 of the esti-
mated PojJulation in England and Wales, excluding casuals and
insane, during the years 1900-1914. The following are the statistics
required for the purpose : —

Table (28). Correlation between Ratio of Indoor and Ratio
OF Outdoor Paupers, each measured per 1000 of the
Population.

Indoor

Outdoor

Indoor

Outdoor

Year.

Paupers —

Paupers —

Year.

Paupers —

Paupers —

Rate per 1000.

Rate per 1000.

Rate per 1000.

Rate per 1000.

1900

6-9

15-8

1908

6-8

15-4

1901

5-8

15-3

1909

71

15-6

1902

60

15-3

1910

7-2

151

1903

6-2

15-4

1911

7-2

141

1904

6-3

154

1912

6-9

11-2

1905

6-6

161

1913

6-7

111

1906

6-8

160

1914

6-4

10-4

1907

6-8

15-6

The coefficient of correlation in this case comes out negative
and = — -15, but it is very small and probably not significant.
If it were, it would imply that as indoor pauperism diminishes
outdoor pauperism increases, and vice versa.

Example (4). — To find the correlation between the Number of
Cattle and the Number of Acres of Permanent Grass-land in the Coal-
Producing Counties of England (1915).

A Government Report was consulted giving the acreage under
crops and grass and the number of live stock in each petty sessional
division in the country, as returned on 4th June 1915, and the
counties included were those which appear in the coal-mining
reports published monthly in the Labour Gazette.

In each county the petty sessional divisions with the greatest
and the least numbers of cattle and of acres of grass-land were
noted, the numbers being written down to the nearest 1000, and,
after a rough examination of the range of these variables from
county to county, suitable class intervals were chosen and a table
of double entrj^ was drawn up, Table (29), Avith an empty square
ready for each possible pair of variables.

CORRELATION — EXAMPLES

121

Table (29). Correlation between the Number of Cattle
AND the Number of Acres of Permanent Grass-land in
the Coal-Producing Counties of England (1915).

Total Head of Cattle (expressed to nearest thousand)

x^

^2

^3

X^

0-5

5-10

10-15

15-20

20-25

25-30

30-35

35-40

Totals

Mean x

: lo

y^

:: '5

15

2-50

0-5

: : 150

^2

; : ,9
! ! :27

4
3

30

3 00

T3

5-10

: : '."(>

: iz

^3

! : : 6
I i -30

I: 3
:: 18

48

4-37

a

CO

o

10-15

I i :i8o

:: 54

4
3

: : : 2
i j :30

33

7-04

So
CO

15-20

; 12

\\\6o

'. '• *

0

: : :25

: 5

30

8-33

20-25

1 : i^S

; 0

0

: °

• 0

0

CO

1

: . 14

: 9

2

26

9-81

25-30

• 0

: : 0

S 0

; 0

-I
: 6

i i 0
: : 22

I
3

31

12-02

30-35

: -6

: : '.°

•: 3

-2

: 0

2

4

1

: 12

: 6

. 4

23

15-33

a

35-40

-2

: : 0

i '^

: 16

CO

0

3

9

i3

3

. 4

1

8

16-87

c5

HO

40-45

: 0

; 12

. 9

0

4

8

16

to

3

3

3

1

10

1900

45-50

; 0

: 12

: 24

. 16

5

10

15

CI.

. 4

. 4

1

9

20-83

•^

50-55

: 20

: 40

. 'S

6

12

24

30

^

1

2

1

1

5

26-50

55-60

. 6

: 24

. 24

. 30

21

1

1

27-50

o

5

"a
»2

60-65

. 21

24
1

1

27-50

65-70

. 24

27
1

1

27-50

70-75

. 27

40
3

3

32-5

75-80

1 120

n

22

^?

1

1

2

20-0

80-85

. "

. 22

Totals

76

97

54

24

14

5

5

1

276

Mean y

9 14

2013

33-24

43-33

50-00

59-50

67-50

57-5

122 STATISTICS

Each petty sessional division was then considered in turn and a
dot was inserted in the particular square applicable to it : e.g. a
petty sessional division with 42,000 acres of grass-land and feeding
19,000 cattle would be represented by a dot in the square defined
by row (40-45) and col. (15-20) in Table (29) ; x was used to repre-
sent the number of cattle and y the number of acres of grass-land
in any division, each expressed to the nearest 1000 units. All the
dots were ultimately added in each square giving the frequency
for each corresponding pair of variables, and these frequencies were
recorded in the centres of the squares to which they applied : e.g.
the frequency of petty sessional divisions stocking 10 to 15 thousand
cattle and with 30 to 35 thousand acres under permanent grass
was 22. The total frequency for each row, i.e. each array of
selected y type, was also noted, in the column at the end of the
rows : e.g. altogether 31 petty sessional divisions were observed of
the type having 30 to 35 thousand acres of land under permanent
grass. Likewise the total frequency for each column, i.e. each
array of selected x type, was noted in the row at the foot of the
columns : e.g. altogether 54 divisions were observed of the type
stocking 10 to 15 thousand head of cattle.

It was possible now to treat each column separately and to
calculate the mean y's associated with different types of x, namely
x^, x^, x^, . . . , and the frequencies so obtained were inserted in
the bottom row of Table (29) : e.g. when x lies between 20 and 25
thousand, the mean value of y is 50 thousand. The resulting
points— (a^i, y^), [x^, y^), {x^, Pa) • • ■ in the notation of Chapter x. —
are plotted together in fig. (25), and they are seen to lie approxi-
mately in a straight line. The successive rows were treated in
precisely the same way and the mean x's calculated corresponding
to y's of cUfferent types, namely y^, y^, y^, . . . , the frequencies
obtained being recorded in the extreme right-hand column of
Table (29) : e.g. when y lies between 45 and 50 thousand, the mean
value of X is 19 thousand. The resulting points {x-^, y^), {x^, y^),
(^3> 2/3). • • • . are also plotted in fig. (25), and, excepting for values
which depend upon only one or two records, they too lie roughly
in a straight line which is not far from coinciding with the previous
one, so that we shall expect on calculation to get a high value for
the coefficient of correlation.

In order to calculate r we need first to find the mean and standard
deviation for each variable. For this let us take as origin the
point (12-5, 27-5). The essential details are shoAvn immediately
below the relative Tables (30) and (31).

CORRELATIOK — EXAMPLES

123

TABiiE (30). Distribution of Petty Sessional Divisions ac-

COKDING to the HeAD OF CaTTLE (EXPRESSED TO NEAREST
1000) STOCKED.

(1) (2) (3) (4) (5)

No. of Cattle

Devia-

No. of Pettv

Product of

Product of

stocked (in

tion from

Sessional

Nos. in

Nos. in

thousands).

12-5.

Divisions.

Cols. (2) .t (.3).

Cols. (2) & (4).

(x)

0-5

_2

76

-152

304

5-10

-1

97

- 97

97

10-15

0

54

. .

15-20

+ 1

24

+ 24

24

20-25

+ 2

14

+ 28

56

2.5-30

+ 3

5

+ 15

45

30-35

+ 4

5

+ 20

80

35-40

+ 5

1

+ 5

25

276

-157

631

Mean number of cattle=12-5— 17^x5=9-66, since x= — wjl class
units referred to 12-5 as origin; and (Tx=5\/[iTw~{^^i)^]
= 5\/l'963=7-00.

[The numbers in col. (4) may be spoken of as the first moments
of the totals of x arrays and the numbers in col. (5) as the second
moments.]

In order to calculate easily the product deviation with reference
to (12-5, 27-5) as origin, the value proper to each square was inserted
just above the frequency and the product of the deviation by the
frequency was inserted just below the frequency in different type of
print to prevent confusion : e.g. the row (50-55) is +5 class intervals
distant from the row (25-30) containing the origin, and the column
(20-25) is +2 class intervals distant from the column (10-15) con-
taining the origin ; hence, for the jDarticular square defined by this
row and this column, the product deviation=5x2=10 ; also
the frequency recorded in this square =4, so that it supplies a
term 10 X 4 to the product deviation ; the numbers 10, 4, and 40
are therefore the numbers Avhich appear in the square. It is neces-
sary to be careful with the signs ; if the product deviation is to
be positive, the separate deviations must be of like sign, both
positive or both negative : hence they must either be both above
or both telow the numbers 12-5 and 27-5 respectively from which

124

STATISTICS

they are measured. In this instance there are only two negative
terms among the product deviations in the whole table.

Table (31). Distribution of Petty Sessional Divisions ac-
cording TO THE Number of Acres of Land (expressed to
nearest 1000) under Permanent Grass.

(1) (2) (3) (4) (5)

143

2T^

Mean number of acres=27-5— ^7^x5=24-91, since y
class units ; and CTj,=5V[W/-(i4|)2]=5\/l04()2= 16-12.

[The numbers in col. (4) are the first moments of the totals of y
arrays, and the numbers in col. (5) are the second moments.']

It is now a simj^le matter to sum the j)roduct deviation terms,
taking each column (or each row) in turn : e.g. the first column
gives

150+216+180+12 = 558 ;

the second column gives

12+54+60+25-6-2=143,
and so on ; and, summing these results together, we get
558+143+76+126+96+160+30=1189.

CORRELATION — EXAMPLES 125

But this is the sum of all the product deviations referred to
(12-5, 27-5) as origin. Transferring now to the mean, we have

118 9 / 157 w 14 3 v

=4-013, expressed in class units.
Hence, r=plax(Ty,

where a^ and (jy are also to be expressed in class units,

=4-013/ V(l-963)-v/(10-402)

-=0-89,

a result not far from unity, so that the correlation is high.

The regression of ' acreage of grassland ' (Y) on ' head of cattle '
(X) is given by

(Y-24-91)=A(X-9-66)

= (0.89)y^\x-9-66),
(7-00)

i.e. Y=2-05X+5-ll.

The points representing the mean y's for x's of different types
should lie close to this line which is shown in fig. (25). This equation
enables us to jaredict the acreage under permanent grass to be
found on the average in petty sessional divisions with a given total
head of cattle in each. The words ' on the average,' to be tacitly
understood even if not stated in all such cases, are emphasised
because the prediction relates to the whole array of divisions of a
particular type, and as it only professes to give the mean or most
likely result it is not to be pronounced worthless if it fails in an
individual trial with a selected division.

Again, the regression of X on Y is given by

(X-9-66)=r^(Y-24-91)

i.e. X=0-39Y+0-05,

which tells us the total head of cattle (X) to be found on the average
in petty sessional divisions when the acreage under permanent
grass (Y) is known. This line is also drawn in fig. (25).

Example (5). — The data for this example are taken from an
exceedingly interesting Government Report on the Cost of Living
of the Working Classes {Rej)ort of an Inquiry by the Board of Trade
into Working Class Rents and Retail Prices together with the Rates

126

STATISTICS

of Wages in certain Occupations in Industrial Towns of the United
Kingdom in 1912 in continuation of a similar Inquiry in 1905.

Y

-_ __ —

^ 7

^

r r

> 7

/ r _

^ llf\

vi\1 / * '

-12)^ „ 7

i '

I—ZaII

F.__ ^

i 1

°t L

I 1

t t _

^7

"53

1 L

t° J ' it

i t

t J

^ ^ _

-s:

i 1

t °L

HJ

4 J- it

JSen-- _- _

t L .

f- J, -

«

1 I

i

' J°

c ,

(

o - it _ _Z

J.- -

•*- T Z^

t

•?s X- itit 4

1

§ i

? it -,

1 It

§ =o - - t

T

^ uO f .

"^

£5. It J L

^ r 1

V* -it

- it _ _ _ t~4 -

•^ It it t t

"= 4' 4

^ tU-

> It

S dO / /

« ^-f-

^ ^2^

c3 p.

•w ^'^Zt

s; __ _ _ I4- _

S -- - - /2^- -H

^ ^v

H ^2

S -.U

S ^o - - -^ Z^ it i

o" /

^ TX ^2^

•^ T i °J

0^ t

to .t 'Kyi

s^ jM

u ^'^ it -

^ -^ i- -|-

i

tni -fines rf^reqr jssioirare-

"+--,„„ -^y Trhe-equdtivnso/

° // /t^ //!

n /Ti V-4 l^ ,11

fe /^ ^ '^

^ yjt -1- t(2 X^

=^0^3'9 y+O-b'5

R -^ ll l^Jl^ }l' f

s: ']'" vmrciT cjypruxrm

ate res pi ctiv'eh~txrlrn~es~

Tl-t-fl' -»/-«■' l>' lltl

""• /^^ iiV Meah^iXs tor

-xls -opdifferent^ type^ < \ [* -
(/ s of cliff ere (ft tc/pe -i-— i-a,

•je .-> 7 ^ (2J /J/er/7^x's /"or

ctratrttreri reanipe6724--91f '

7 •■f We^lii esn'ntqrse

' h of the distiit iiti

oil

. \ .

1 I

'

r it

i °

0 10 20 30 40 X

Total Head of Cattle (expressed to nearest thousand)
Fio. (25).

Cd. 6955). Some further particulars concerning this Report will
be found on p. 281.

CORRELATION — EXAMPLES 127

The towTis included in the inquiry' numbered 93, but in five
instances it was found desirable to consider closely adjacent muni-
cipalities as single toAvus thus reducing the number of to"WTi-units
to 88, namely 72 in England, 10 in Scotland, and 6 in Ireland. In
the example which follows the three zones of London, middle,
inner, and outer, have been treated as separate towns, so making
the net number of town- units 90. This number is too small to
allow any real value to be attached to our results, but the fewTiess
of the observations makes them easier to deal with as an illustration
of method.

We begin as before by choosing convenient class intervals for
the two factors we propose to consider, namely. Increment of Un-
skilled Wages and Increment of Rents — by increment in each case
is meant the percentage increase (+) or decrease (— ) between
1905 and 1912 — and then form a correlation table. In the last
example separate tables were drawTi up to find means and S.D.'s,
but that was only done in order to keep the argument clear at its
first presentment : generally we may dispense with these additional
tables and show all the working in one (see Table (32)).

The increment of wages runs from (— 2-5) per cent, to (+11-5)
per cent., so that, if we take (—0-5) as origin and a difference of
2 per cent, as miit, the classes run from (—1) to (+6), these numbers
being sho^vn in different type in the table, but in the same com-
partments as the others. In the fourth row from the bottom
are shown the total frequencies for x arrays from class (—1) to
class (+6), and in the row just below it these several frequencies
are shown multiplied by their corresponding deviations measm-ed
from (—0-5) as origin in terms of the class unit — the resulting
numbers give the first moments of the totals of x arrays. These
numbers, multipHed again by their corresponding deviations, give
the second moments of the totals of x arrays, and appear in the
last row but one of the table.

We deal in exactly the same way with increment of rents : a
percentage increment of ( — 1) is taken as origin from which devia-
tions are measured, a difference of 3 per cent, is taken as unit,
and the different classes then have deviations running from (—3)
to (+6). The totals of y arrays, the first moments, and the
second moments of these totals appear in the last three columns
on the right-hand side of Table (32).

To calculate the deviation products, numbers were inserted in
each square on the same principle as in the last example, and the
sums of these products for each x array, that is for each column,

128

STATISTICS

are given in the bottom row of the table — 1, 0, 14, 6, etc., making
in all a total of 126.

Table (32). Cokeelation between Increment of Unskilled
Wages and Increment of Rents in certain Industrial
Towns of the United Kjngdom.

X = Percentage Increment of Wages

-I

-2-5

0

-0-5

f 1
1-5

+ 2.

3-5

+ 3
5-5

+ 4
7-5

+ 5
9-5

+ 6

11-5

Totals
of y
arrays

1st.
mo-
ments

ofy
arrays

2nd.

mo-
ments

ofy
arra'is

CO

-3

-10

o
1
o

1

-3

9

eg

CO

CO

-2

-7

2
1

2

o
3
o

4

-8

\6

CD

-I

-4

o
4
o

-I
1
-I

-2

2

-4

-4

1
-4

-5

1
-5

-6
1
-6

10

-10

10

O

-1

o

15
o

o
1
o

o
6

0

0

6
o

0
2

0

30

-

-

+ 1

2

-I

1

-I

0

9
o

I

1

I

2

3

6'

4
3

12

5

1
5

18

18

18

i

+ 2

5

0

6

o

2
1
Z

4

1
4

8
1
8

10
2
20

11

22

44

+ 3

8

o
3

o

3

4

12

9

1
9

12

1
12

IS
2
30

11

33

99

+ 4

11

o
3
o

3

12

48

1

■+S

14

o

1

o

1

5

25

ih

+ 6

17

24

1
24

1

6

36

Totals of X arrays

2

45

8

12

7

7

8

1

SO

75

305

7st. moments of
X arrays

-2

-

8

24

21

28

40

6

125

2ncL moments of
X arrays

2

-

8

48

63

112

200

36

469

Product Sums of
X arrays

1

0

14

6

9

52

50

-6

126

Total Product Sum

The necessary calculations are as follows : —

1. Mean a;=-0-5+2(125)/90=2-28,

a^=2v'[-*9V-(-V(r)2]=2V(26585)/90.

2. Mean ?/=-l + 3(75)/90=- 1-50,

<^v=3V[%V-(H)2]=3V(21825)/90.

3. p=V/— (V(f)(if)= , expressed in class units.

(90)

CORRELATION EXAMPLES

129

Hence

r=pla^ay

1965

X

90

X

90

(90)2 y^(26585) V(21825)
=0-08.

In substituting for o-j. and ay to find r we have omitted the factors
2 and 3 respectively, because the S.D.'s have to be expressed in
the same units as p. Alternatively, if we worked Avith a difference
of 1 per cent, as unit, instead of taking a difference of 2 per cent,
as unit for x deviations, and a difference of 3 per cent, as unit for
y deviations, each individual product of x and y deviations would

Y

1'

\ ~

■

-1

~

-r.

(■'

■1

6

-

-

-

<o

c

cgs

«<-,

•4.J

-

-

-

-

§4

-

s:
"~3

&

c

c

p2

(J

l)

-

_

n

CJ

-

^

_

— 1

Q,

-

_,

__

»"

■

I

vl(2-.

^8,!

■b)

"^

r-

-

^

_

_

_

_

_

_

_

_

__

O 12 3 4 5

Percentage Increment of Wages
Fig. (26).

have to be multiplied by 2 x 3. Thus p would then be 6 X 1965/(90)2,
and we should get the same result for r as before by taking 0^.
and ffj, as in (1) and (2) above. In this case r is so small as to be
quite insignificant of any correlation between the two factors dis-
cussed, and the regression lines should therefore be not far from
perpendicular to one another.

The regression of y on x, or the equation giving the most probable
y for a given type x is

(t/-l-50)=r^(a;-2-28),

i.e. t/=0-lIa;+l-25.

I

130

STATISTICS

Similarly, the regression of x on y is

.T=0-062/+2-2.
To draw the first line we note that it passes through the points
(0, 1-25) and (5, 1-8) ; also the second line goes through the points
(2-2, 0) and (2-5, 5). The two lines intersect at M (2-28, 1-5), the
mean of the distribution. They are drawn together in fig. (26).

Table (33). Correlation between Unskilled Wages
AND Rents in certain Industrial Towns of the
United Kingdom.

x= Index Number for Wages of Unskilled Labour

45-5

51-5

57-5

63-5

69-5

75-5

81-5

87-5

93-5

99-5

CO

40-5

2

3

1

(3

48-5

2

1

3

1

4

3

1

0:1
"0

56-5

1

1

2

7

15

6

2

64-5

2

1

3

9

4

1

72-5

1

3

3

2

42

s:

80-5

1

1

1

1..

88-5

1

.0

s

1

96-5

1

104-5

11

112-5

1

Example (6). — Instead of discussing the Changes in Wages and
Rents between 1905 and 1912, it might be of interest to find the
correlation between index numbers representing Actual Wages and
Rents in October 1912, taken from the same Report. The necessary
data for this purpose appear in Table (33) showing the distribution
of frequency between the different classes : e.g. seven towns were
observed in which the index number for wages was between the
limits (79-84) and the index number for rents was between the
limits (53-60). The wages figures quoted in Table (33) refer only
to unskilled labour in the building trade ; the inquiry actually
embraced certain occupations in the building, engineering, and

CORRELATION — EXAMPLES

131

printing trades, these having been selected as industries which are
found in most industrial towns, and in which the time rates of
wages are largely standardised.

Table (34). Correlation between Increment of Working
Class Prices and Increment of Working Class Rents
IN certain Industrial Towns of the United Kingdom.

X = Percentage Increment of Prices

7-5

9-5

11-5

13-5

15-5

17-5

19-5

-10

1

-7

1

1

2

to

-4

1

2

2

2

1

2

0=

-1

1

4

6

10

8

1

s:

2

1

2

5

8

1

1

to

5

2

4

2

3

8

1

2

3

1

4

II

Si

11

1

1

1

14

1

17

1

The coefficient of correlation turns out to be 0-46, distinctly larger
than in the previous case. Also the lines of regression are : —
(1) ?/=0-47a;+21. (2) a;=0-45?/-f 56.

Example (7). — The Report also furnishes data for evaluating the
correlation between the Increment of Working Class Prices and
Increment o/ Working Class Bents, again meaning by increment the
percentage increase (+) or decrease (— ) between 1905 and 1912
(see Table (34)).

The correlation in this case is very small, being only 0-13. The
regression equations are : —

(1) ?/=0-22a;-l-5. (2) x^0-01y-\-l3.

PART II
CHAPTER XII

INTRODUCTION TO PROBABILITY AND SAMPLING

Suppose we wish to know the average measurement of some organ
or character, e.g. length of forearm or weight or anything similar,
in a large population containing several thousand individuals. The
mean obtained by actual measurement if it were practicable to
carry it out on so large a scale, would evidently depend to some
extent upon the sex, the race, the age, the social class, and so on,
of the individuals selected, and we shall accordingly assume our
population to be composed of individuals of the same race and sex,
at about the same age, taken from the same class, etc. ; it would be
impossible in practice no doubt to secure that all conditions should
be identically the same for all the individuals observed, but the
population may be as homogeneous as we care to make it in theory.

Now suppose that, instead of attempting to measure every single
individual, a random sample of 1000 from among the population
be taken and that the mean and variabihty of the measurements
for this sample be calculated, giving results m^ and a^. With
these may be compared Wg and a^, the results of measuring a second
sample of 1000 individuals, m^ and a^, the results of a third sample,
and so on. It is extremely unUkely that the values obtained for
the 771 's in this way will equal one another, neither will the ct's
be equal ; but, if we have succeeded at the beginning in avoiding
all ill-balanced influences when we tried to make the field of
observation as homogeneous as possible, the resulting m's and a's
will only differ from the values of the mean and variability for the
whole population, assuming they could be measured, within a
comparatively small range.

Differences of this kind, which arise merely owing to the fact
that we are often obliged in practice, for lack of time or means, to
deal with a comparatively small sample instead of with the whole
population of which it forms a part, are said to be due to random

132

INTRODUCTION TO PROBABILITY AND SAMPLING 133

sampling. Granted that the samples themselves are adequate in
size (containing, say, from 500 to 1000 individuals each) an esti-
mate of differences to be expected between one and another can be
made, and unless the observed differences fall outside recognized
limits it is said that they are not significant of any difference other
than such as might quite well be accounted for by random samphng
alone.

In theory, then, we can imagine a large number of such random
samples selected, and by determining the S.D. of their means,
m^, m^, mg, . . . , we should have a fair measure of the deviation
which might quite well occur from the true value, that is, from the
mean of the population as a whole, through working only with a
sample. Further, a range of two or three times the S.D. on either
side of the true mean ought to take in the majority of the sample
means observed.

Exactly the same principle holds good in dealing with the pro-
portion of individuals in a given population which can be assigned
to a particular class, or in discussing the S.D. of the distribution, or
the C. of v., or a coefficient of correlation, or any other statistical
constant, no matter what the nature of the character may be which
is measured or observed, or whether it relates to animate or inani-
mate objects. Take, for instance, the variability — by selecting
several samples from a given population we get a series of values
<^i. o'2' o's • • •) and in the S.D. of this distribution of variabilities
we have a measure to which we can compare the deviation of any
sample variability, cty, from the true variability of the whole popu-
lation, while a range two or three times the S.D. might be expected
to include the majority of the different variabilities met with in
the samples.

Although the S.D., as we have explained, provides quite a suit-
able measure of the extent of deviation of a sample constant from
its true value in the population as a whole, in practice, owing to
the historical development of the theory having followed the track
of the normal curve of error [see Chapter xviii.] a measure known
as the probable error and equal roughly to two-thirds of the S.D.
is not seldom employed in its place. The main, if not the sole,
justification for retaining this measure is that it has established its
position by long usage, and in any case it is very easily deduced
from the S.D. by the relation

p.e.= 0-6745 S.D.,

which follows at once from the normal curve and is only strictly

134 STATISTICS

justified when the distribution is normal (see p. 246). Let it suffice
here that instead of simply using the S.D., as might now seem
the obvious course, some writers prefer to multiply the S.D. by a
certain fraction, in which there is no particular virtue except that
which arises through honourable descent, and to work with the
' probable error.'

Since we do not know how much v/eight to assign to any result
unless the magnitude of its p.e. is also given, results are frequently
stated in the following manner : in a study of the Variation and
Correlation in the Earthworm, by R. Pearl and W. N. Fuller [Bio-
metrika, vol. iv. pp. 213-229] :--

Mean length of worm=19-171±0-094 cms.,

S.D. = 3-077±0-067 cms.,

C. of V.=16-049±0-35G per cent.,

meaning that the mean length of the Avornis measured Vvas 19-171
cms., subject to a probable error of 0-094 cms. which might be in
excess or defect, in other words the mean length lay probably some-
where between

19-077 cms. and 19-265 cms. ;

similar remarks apply to the variabihty, absolute (S.D.) or relative
(C. of v.).

When the standard deviation (p.e./0-G745) is used as the measure
of error due to simple sampUng, the fact is generally recorded, and
it is sometimes spoken of as the standard error in that connection,
but, as it seems unnecessary to multiply names for ideas which are
not really new, only that they appear in a new setting, we shall
not employ the term.

It must be clearly understood that no outstanding and predict-
able cause exists, by our hypothesis, for such differences as occur
in the statistical constants between one sample and another : they
are the resultant effect of a complex of forces which cannot be
properly traced, still less measured, apart from one another, and
which have been happily described as that ' mass of floating causes
generally known as chance.' Since therefore the forces coming
into play, under the ideal conditions formulated, are of the same
chance nature as those affecting the spin of a well-balanced coin
or the selection of a card from a smooth and well-shuffled pack,
it may be expected that the resulting distribution of means,
Wj, m^, mg, . . . , of S.D.'s, o-^, ag, as, ... , and of all the other
constants will likewise be subject to the same laws of probabiUty

INTRODUCTION TO PROBABILITY AND SAMPLING 135

which serve to describe within limits what happens in the case of
coin or card. It follows that some acquaintance with the first
elements of mathematical probability is essential if one is to under-
stand the theory of sampling, and a short digression must here
be made in order to introduce that subject. This will be found
to lead directly to a solution, under certain prescribed conditions,
in the simple case when the character observed is an attribute like
complexion, fair or dark, or like birth, male or female, which can
only fall into one of two definite classes and when every one observa-
tion in the sample is independent of every other. In the more
general case where the character observed is capable of direct
measurement and may lie in. magnitude anywhere along a scale
of values divided up into a number of different classes, it is not
so easy to determine the effect of random sampling, because it is
not possible, as it is in the previous case, actually to draw up a
frequency table describing in detail the character of the distribu-
tion to be expected from theory in any given sample.

The idea contained in the word probability is one famiHar to us
in our everyday talk, but if we seek to analyse it as used we find
it as elusive as the personality of the user. A remarks : ' Wars
will probably be stamped out, like duelling, in the course of time.'
B replies : ' No ! fighting will probably go on as long as the world
lasts — you can't change human nature.' Now the amount of
credence we are prepared to give to each of these statements is
vague and uncertain until we know something about A and B
themselves and the value of their judgment, quite apart from the
influence of our own opinion upon the matter ; perhaps A is an
optimist or B is a pessimist, and in estimating the ' probably *
used by each we must allow for these facts. Probability, then, in
ordinary conversation, is something largely subjective : it has a
varying significance according to the person who uses the word
and, unless we could get rid of this personal element, it would be
hopeless to try and approach it along scientific lines.

Mathematical probability is unlike colloquial probability in that
all the uncertainty is taken out of it, or at least the uncertainty is
confined within defined limits. We shall only touch the fringe of
the subject in this book, and what we have to say may be best
introduced by considering some examples which may appear trivial,
but they possess the merit that no personal bias can enter into
their discussion to distort the results. The reader must not be
impatient at their artificial character : in many, if not in all,
branches of science, before tackling any particular problem as it

136 STATISTICS

actually exists, it is helpful to examine what can be deduced in a
simple case free from all complication, and, having settled that,
we try to see how the results are affected when we come to allow
one by one for the various complicating factors which exist. For
example, in Astronomy, the track of a planet in space may first be
found on the hypothesis that the sun alone is the compelling influence.
Then we may proceed to discuss how it is deflected from its path
when the gravitational influence of neighbouring planets also is
taken into account.

Let us start with an ordinary pack of playing cards, and, after
shuffling, turn up one card. Can we measure the probability that
this card shall be (1) the 7 of spades ? (2) some spade ?

Altogether there are 52 cards, and we will suppose that the
cards are so cut and so smooth that each of the 52 has an equal
chance of being turned up : for instance, there is to be no sticki-
ness or anything to help any particular card to evade us by sticking
fast to its neighbour. Now we are certain to turn up some card
and there are 52 different possibilities, each of them by hypothesis
equally probable. If, then, we agree to denote certainty by unity,
we must divide 1 into 52 equal parts and assign one part to each
card as the probability of its appearance.

1. The probability (or chance as it is sometimes called) of .turning
up any stated card, such as the 7 of spades, is therefore 1 out of 52,
i.e. 1/52.

2. Again, since there are 13 spades in all, the chance of turning
up some spade is 13 out of 52, i.e. 13/52=1/4.

These results may be put in another way which is often useful.
If the experiment is repeated a great number of times, a return to
the initial conditions of the problem being made after each trial
by replacing the card drawn and reshuffling the pack, we should
expect to turn up the 7 of spades on the average about once in
every 52 experiments, and we shoiild expect to turn up some spade
on the average about once in every 4 experiments. This must
not be taken to mean that in 4 experiments we are sure to turn
up just one spade — a trial will readily prove such a statement to
be untrue— but that, if we went on performing experiment after
experiment, we should in the long run get a proportion of about
1 spade to every 4 experiments and a trial will likewise prove the
truth of this statement.

Generally, when an event can happen in n different ways alto-
gether, and among these different ways there are a which give
what might be called successful events, the probability of success

INTRODUCTION TO PROBABILITY AND SAMPLING 137

at any single happening is a out of n, i.e. a/n, and is usually denoted
by the letter p, and the probability of failure is {n—a) out of n,
i.e. {n—a)/n, and is usually denoted by the letter q.

Clearly {p-\-q)^l, and this is reasonable because we are certain
to get either a success or a failure at a single trial and unity was
fixed as the measure of certainty. In k trials, the probable number
of successes would be kp and of failures kq, because in n trials, on
the average, there are a, or np, successes and (n—a), or nq, failures.

Examj^le (1). — In the second case considered above, the pro-
bability of success (turning up a spade) is a out of n

-:a/w= 13/52= l/4=p,

and the projability of failure (not turning up a spade, i.e. turning
up one of 39 other cards) is (n—a) out of n

= {n-a)/n=d9/52=3/4:=q.
And (2>+g)= 1/4+3/4=1.

Example (2). — What is the chance of drawing either a picture
card or an ace from the pack at a single trial ?

Altogether there are 12 picture cards, and the chance of drawing
any one of them is thus 12 out of 52

= 12/52=3/13;

and the chance of drawing any one of the 4 aces is 4 out of 52
=4/52=1/13.

Hence the total probability required

= 3/13+1/13=4/13.

Generally, if the jDrobability of one type of event is p^, and the
probability of a second type of event is p^, and if either type is
reckoned a success, then the total probability of success is (^1+^2)*
This evidently holds good however many different types there
may be, and even if there is only one event of each type.

Consider now the simultaneous happening of two events, one of
which can happen in n different ways, a among which are to be
regarded as successful, and the second can happen in n' different
ways, a' among which are to be regarded as successful. Further,
the two events are to be absolutely independent of one another
in the sense that neither is to influence the success or failure of
the other. What is the probability of a double success occurring ?

The total number of different combinations of the two events

138 STATISTICS

possible is nn' , for any one of the n possible happenings for the
first event can be combined with any one of the n' possible happen-
ings for the second event. Also the total number of different
combinations of two successes possible is aa' , for any one of the
a possible successes for the first event can be combined with any
one of the a' possible successes for the second event. Hence,
according to our definition of probability, the probability of a double
success is aa' out of nn'=aa'/nn'={al7i){a'/n').

Thus to get the probabiHty of a double success for a combination
of two independent events we must multiply together the separate
probabilities for the success of each event taken by itself.

Similarly, in the above case, the probability of a double failure
= {n—a){n' —a')lnn' ; and the probability of one success and one
failure

a n'—a'n~a a'
n n n n

for the first event can be a success and the second a failure or the
first a failure and the second a success.

Here, again, if we take all the different possibilities into account,
and add the probabilities corresponding to each case, we arrive
at certainty, the measure of which is unity, thus : —

probabiHty of 2 successes —aa'/nn',

„ 1 success and 1 iai[uTe=a{n' —a')Jnn' -\-a' {n—a)/nn'

„ 2 failures ={;n—a){n'—a')lnn'.

Therefore total probability, all cases,

aa' a{n'—a') , a'{n—a) {n—a){n'—a')

= — ,+ -, — -r- — -r ;

nn nn nn nn

— {aa'-\-an'—aa'-\-a'n—a'a-\-n7i'—na'—an'-\-aa')lnn'

=nn'Jnn'

= 1.

Example. — Take two packs of cards. What is the probability
of drawing an ace from the first pack and a king, queen, or knave
from the second pack ?

Here a=4, n=52, a'=12, n'=o2 ; hence the required probability

=aa7wi'=4/52x 12/52=3/169= 1/56|.

Thus we might expect to succeed on the average about once in
56 trials.

INTRODUCTION TO PROBABILITY AND SAMPLING 139

We proceed to discuss the case of a coin spun a number of times
in succession, and we shall find the probabilities of the appearance
of so many heads (H) and so many tails (T) in so many spins on the
hypothesis that the coin is perfectly balanced and equally likely
to fall on either side.

In 1 spin there are 2 possible events, namely H or T, which
we shall write simply as

(H, T).

In 2 spins there are 4 possible events, because we can combine
the H or T of the first with an H or T at the second spin, and we
may express the result thus

(H, T)(H, T)=(HH, HT, TH, TT) ;

the interpretation of which is that we may get either head followed
by head, or head followed by tail, or tail followed by head, or tail
followed by tail.

In 3 spins there are 8 possible events, because we can combine
the 4 events previously possible with an H or T at the third spin,
thus getting

(H, T)(H, T)(H, T)

= (H, T)(HH, HT, TH, TT)

= (HHH, HHT, HTH, HTT, THH, THT, TTH, TTT) ;

the interpretation of which is that A\e may get either 3 heads in
succession, or 2 heads followed by 1 tail, or head followed by tail
followed by head, and so on.

In 4 spins there are 16 possible events, because we can combine
the 8 events previously possible with an H or T at the fourth spin,
thus

(H, T)(HHH, HHT, HTH, HTT, THH, THT, TTH. TTT)
= (HHHH, HHHT, HHTH, HHTT, HTHH, HTHT,
HTTH, HTTT, THHH, THHT, THTH, THTT,
TTHH, TTHT, TTTH, TTTT).

But the method here adopted to get the possible events at each
stage is precisely the same as that which gives the successive terms
in the ordinary algebraical expansions of

(H+T), (H+T)(H+T), (H+T)(H+T)(H+T), etc.
Also each new spin has the effect of doubling the number of possible

140 STATISTICS

events obtained at the previous spin, and we conclude that in
n spins, there are

(2x2x2x . . . to w factors),

or 2", possible events, and these events are given by the successive
terms in the expansion of

[(H+T)(H+T)(HH-T) ... to w factors.]

Let us now consider the probabilities of the different events
obtainable. The important point to notice is that at any stage
each possible event has exactly the same probability, for there is
no reason why any particular spin should give H rather than T,
or T rather than H : for example, in 3 spins there are 8 possible
events, each by itself equally probable, and we therefore divide
the unity of certainty into 8 equal parts and assign one part to each
event, thus

probability of 3 heads— HHH=|
probabihty of 2 heads and 1 tail— HHT=|^

HTH=i i
THH=iJ
probability of 1 head and 2 tails— HTT=J)

THT=| i
TTH=iJ
probabihty of 3 tails— TTT=J.

It is clear from this arrangement that, if the order of the appear-
ance of H and T is indifferent, some events are of the same type
and some types are likely to appear oftener than others, e.g. the
probability of getting ' 2 heads and 1 tail ' (or ' 1 head and 2 tails ')
is three times as great as the probability of getting ' 3 heads '
or ' 3 tails.' Hence for conciseness it is convenient to adopt the
ordinary index notation and write

HHH=H3, HHT=H2T, HTH=H2T, etc.,
so that the possible events in 3 spins are

W, 3H2T, 3HT2, T3 ;
in 4 spins they are

W, 4H3T, 6H2T2, 4HT3, T* ;
and so on.

The probability of any jDarticular type is now readily written
down : e.g. in 4 spins, the probability of getting 2 heads and 2 tails

= (number of successful events possible)/(total number of events

possible)
=6/2*=6/16=i.

INTRODUCTION TO PROBABILITY AND SAMPLING 141

But the binomial expansion always sums together terms of the
same type for us in just the manner wanted, and we have the
possible events in n spins given by the successive terms in the
expansion of

(H+T)(H+T)(H+T) ... to 71 factors,
i.e. (H+T)«,

i.e. H«+"Ci . H«-iTi+«C2H«-2T2+ . . . +T«,

and therefore again the probability of any particular combination
is readily written down : e.g. probabiUty of ' (w— 2) heads, 2 tails '

= (number of successful events possible)/(total number of events
possible)

= "C2/2«.

Another way of stating the result obtained is to say that we
might expect to get

n heads appearing on the average about once in every 2'* trials,
(n—1) heads, 1 tail „ „ ,, "Cj times „

(7i— 2) heads, 2 tails ,, „ ,, "Cg times „

and so on.

K, in accord with our previous notation, we call the appearance
of, say, H at any spin a ' success,' and label its probability ^ by the
letter p, and if consequently the appearance of T at any spin is a
' failure,' its probability, |, to be labelled by the letter q, we have the
probabilities of the different combinations of events in (H+T)", or
H«+«CiH«-iTi+"C2H"-2T2+ . . . +T«,

given by the corresponding terms in (p+g)", or

where p=q=h

After each spin of the coin in the case considered the distribution
of probabilities was symmetrical, e.g. after the fourth spin the pro-
babilities were

1 _* JL _* J-

lF» iB^' 16> iFl l^"'

We pass on now to a case where the distribution is not symmetrical,
owing to the fact that p and q are no longer equal for any isolated
event.

Consider the throw of an ordinary die in which each of the six
faces is assumed to have an equal chance of appearing uppermost.
The probability of throwing, say, a 3 is 1/6, since we are certain
to throw either 1, 2, 3, 4, 5, or 6 ; and the probabihty of failing to
throw a 3 is 5/6, since we are certain either to throw a 3 or not
to throw a 3.

142

STATISTICS

If we represent the probability of success (say, in this case,
throwing a, 3) hy p {i.e. 1/6), and failure {i.e. in this case, failing
to throw a 3) by g {i.e. 5/6), we have

^+5=1/6+5/6=1.
Bearing in mind then that the probability for a combination of two
independent events is determined by multiplying together the
separate probabilities for each, we have the following table shoeing
what might be expected when 1, 2, or 3 dice are thrown up together,
where s stands for success and / for failure : —

No. of

Dice

thrown.

Different
Possibilities.

Different
Probabilities.

1
2

3

ss, sf.

fs,ff.
sss, ssf, sfs, sff,
fss,fsfjfs,fff.

pp,M-

qp, qq.
ppp, ppq, pqp, pqq,
qpp, qpg, qqp, qqq.

The table is easily extended on the same principle, and at each
step, it will be noticed, a fresh pair of possibilities, s or /, is intro-
duced, with corresponding p or q, to be combined with what has
gone before.

If the order of appearance of 6- and / is a matter of indifference,
e.g. if it does not matter whether the first die shows s and the
second /, or vice versa, so that results of the type sff and fsf may
be regarded as equivalent, we may use the index notation, as in
the coin case, to render the table more concise, thus : —

No. of i

Dice !
thrown. ,

Different
Possibilities.

Corresponding
Probabilities.

1 s, f. p, q.

2 s\2sf,p. I p\2pq,q^.

3 S\ 3S2/, 35/2, p^ p3^ 3p2^^ 3p^2^ g3

When, therefore, n dice are thrown we again recognize the
different possibiUties as given by the successive terms in the ex-
pansion of (5+/)'*, namely

s«+«CiS"-^f+"C2.s«-^/'^+ ...+/«,
and the corresponding probabilities by the successive terms in the
expansion of (p+g')", namely

p''+«Ci2)'»-Y+"C2P«-V+ • • • +9''*-

INTRODUCTION TO PROBABILITY AND SAMPLING 143

Hence the probability of throwing n threes =2)** =1/6" ;

(n-1)

(n-2)

_ 1 ^

~^' 6^^ '6

=5ri/6" ;

n{n—l)

1

1-2

=25w(w-l)/2.6";
and so on.

The result we have just obtained is of perfectly general applica-
tion. Whether we spin 7i coins, in which the probability, p,^ of
success (say * heads ') for each is 1/2, or throw n dice, in which the
probability, p, of success (say ' to get a 3 ') for each is 1/6, or have
any n similar but independent events happening in which the
probabiUty of success for each is p, the different resulting possi-
bihties as to success are given by the successive terms in the expan-
sion of (s+/)"j and their corresponding probabilities are given by
the successive terms in the expansion of {p-\-q)^.

We are thus in a position to form a frequency table, Uke that on
p. 53, showing the probabiUties of getting 0, 1, 2 ... % successes
(in other words, the proportional frequencies of these different
numbers of successes) at the occurrence of n similar independent
events, where p is the probabiUty of success for each and q is the
probability of failure : —

(1)

Table (35). Binomial Distribution.

(2) (3)

(4)

Number of
Successes.

(X)

0
1

Frequency.

(/)
q-

n(n-l) .^

1-2 ^ ^

n(» -!)(«- 2)^ _ 3^3

1-2-3

Product of Nos. in
Cols. (1) & (2).

ifx)
0

Product of Nos. in
Cols. (1) & (3).

1-2

np"

np

0

2n(»-l)g"-V^

3n{n-l){n-2) , ,
F2 ^ ^

n^p"

np[l+p{n-l)]

144 STATISTICS

Col. (1) gives the deviations from the origin of measurement,
which in this case is taken as ' no successes,' the class interval
being equal to a difference of 1 in the number of successes.

The summations of the last three columns are effected as
follows : —

Col. (2). g"+ g-V+^Y7^V-V+ . . . +2>"

= 1,

because p-]-q=l.

Col. (3).

=np.

Col. (4).

=wjp[l+^(w— 1)].

The arithmetic mean of the distribution

=sum of terms in co}. (3)/sum of terms in col (2)

=np.

INTRODUCTION TO PROBABILITY AND SAMPLING 145

The mean-square deviation referred to zero as origin, zero in this
case corresponding to ' no successes '

=sum of terms in col. (4)/sum of terms in col. (2)

=np[l-^p{n—l)].

Thus the standard deviation, a, is given by

a^=np[l-\-p{n—l)]—x^,

where x is the deviation of the mean from the origin of measure-
ment, so that x=^np.

Therefore a^=np[l-\-p{n—l)]—n^p^

=np{l—p)-\-n^p^—ri^p^

=npq.
Hence ct='\/(^P4)»

and p.e.= 0-6745 A/(npq).

These two results are exceeding^ important, and it is essential
to understand what it is they measure. An example may help
to make this clear.

If we spin 300 coins, counting ' head ' for each a success, the
number of heads we shall get will be unlikely to differ very greatly
from the average or mean number of successes, np, i.e. 150 if p=l/2
for each coin, and in the long run, if we repeat the experiment a
great number of times, we shall get a proportion of about 150 heads
to every one experiment. Again, if we throw 300 dice, counting
every throw of the number 5, say, for each die a success, so that
p in this case=l/6, the number of fives we shall get will be unlikely
to differ much from np, i.e. 50, and in the long run, if we repeat the
experiment a great number of times, we shall get on the average
a proportion of about 50 fives to every experiment ; we should
find, for example, something like 5000 fives if we threw 300 dice
100 times in succession. The a.rithmetic mean of the distribution
tells us therefore about what number of successes to expect in one
experiment with n events if n is fairly large, though we should be
unlikely to get exactly this number if we confined ourselves to the
one experiment.

The second result, the S.D., supplies us with a measure of the
unlikelihood of getting the exact number of successes expected at
any single experiment, for it defines the dispersion of the different
numbers of possible successes about their average. Clearly the
greater the dispersion, the greater is the likelihood of missing the

K

146 STATISTICS

average. The mean number of successes when an experiment is
repeated a great number of times is np, but at any single experi-
ment it is not unlikely that the number of successes obtained may
dififer from np by as much as 0-6745 \/{npq) in excess or in defect ;
it is, however, unlikely, as we shall see later (p. 244), that the
number will differ from np by more than S\^{npq) in excess or
defect when the distribution is not very skew, or unsymmetrical,
especially if n be large. The probable error in the case above when
we throw a sample of 300 dice is

=0-6745 V(300x 1/6 x5/6)=0-6745\/(41-67)=4-4,

and it is therefore quite likely that the number of fives obtained
at one experiment will differ from the expected number, 50, by as
much as 4 or 5 in excess or defect, but it is unlikely that the number
will fall outside the limits 50i:3v/(41-67), say 30 to 70.

It is sometimes more convenient to refer to the proportion of
successes, etc., expected at any experiment rather than to the
actual number expected. In that case, since with n events the
expected number of successes is pn, but the number obtained may
quite likely differ from this by ±0-6745-v/(npg'), therefore with
n events the expected proportion of successes is pn/n, i.e. p, with
quite possibly an error=±0-6745\/(wpg)/7i, i.e. :^0-Q74i5\/{pq/n).

Thus, with the 300 dice, the expected proportion of successes at
one experiment lies between

[l/6-0-6745V(l/6x5/6-^300)] and [1/6+0-6745 v/(l/6x 5/6^300)]

i.e. (1/6-0-6745/46-5) and (1/6+0-6745/46-5)

i.e. 1/5-5 and 1/6-6 ;

and it is unlikely that the proportion wOl differ from 1/6 by more
than 3/46-5, i.e. 1/15-5.

To illustrate how the binomial distribution might be directly
applied, an experiment was made with 900 digits selected at random
by taldng in succession the digits in the seventh decimal place in
the logarithms of the following numbers : —

10054, 10154, 10254, . . . 99954,

as given in Chambers's Mathematical Tables. In this way each of
the 10 digits, 0, 1, 2, 3 ... 9, may be supposed to have stood an
equal chance of selection each time one was \M"itten down. Gaps
of 100 were left between the numbers selected so as to avoid runs

INTRODUCTION TO PROBABILITY AND SAMPLING 147

of the same figure which sometimes occur even in the seventh
decimal place owing to lack of independence.

The digits w.ere arranged in 36 columns, each column containing
25 digits, and in this way we obtained what was equivalent to
36 separate but like experiments with 25 events each. If we agree
to regard the appearance of a 7 or an 8 as a successful event, and
the appearance of any other digit as a failure, the chance of success
at any appearance is 2/10, and the chance of failure is 8/10. The
case is thus of exactly the same kind as that of throwing 25 dice
36 times in succession, and if the probability of success, namely 1/5,
for each independent event, be denoted by p, and the probability
of failure, namely 4/5, by q, the distribution of successes and failures
should approximately conform to that given by the expansion of

for any particular experiment, and since the experiment was re-
peated 36 times, the total numbers of successes and failures of
different orders obtained should approximately conform to

36(^+g)25,

for if the probability of an event is p the number of events to be
expected in N trials is Np.

The actual distribution observed is compared with that given
by the binomial expansion in Table (36). Col. (2) is obtained by
picking out the appropriate terms in the expansion of 36{p-\-q)^'^,
where p—l/5, gf=4/5 ; this expansion is

/ 25 25-24- \

36^=^5+^. p2V+y7|^^""3'+ . . . +q''j.

Thus, 5 successes occur

36 ' — - »5«2o

1 • 2 • 3 . . .20

times, and this equals 7 06, or approximately 7.

The mean number of successes by theory=w^=25/5=5. The
mean by trial, since it is measured from zero as origin, the numbers
in col. (1) being the deviations,

=2'(/a;)/i;(/)- 162/36=4-5.

The standard deviation by theory

= V(«P?)=V(25xixl)=2.

148

STATISTICS

Table (36). Distribution of Stjccesses (getting a 7 or 8) in
THE Random Choice of 25 digits 36 times in succession.

(1) (2) (3) (4) (5)

No. of
Successes.

Frequency

Frequencv

Product of

Product of

by

Calculation.

by

Experiment.

Nos. in
Cols.(l)&(3).

Nos. in
Cols.(l)&(4).

{X)

(/)

(/•^)

ifx')

1

1

1

1

1

2

3

5

10

20

3

5

5

15

45

4

7

7

28

112

5

7

9

45

225

6

6

4

24

144

7

4

3

21

147

8

2

0

0

0

9

1

2

18

162

36

36

162

856

By trial, the mean square deviation, measured from zero as origin

==856/36.
Thus the S.D. by trial= V(-V/-«'),
where x is the deviation of the mean from the origin,

= ^[856/36- (4-5)2
= 1-88.

It will be seen that not one of the 36 experiments gave a number
of successes differing from 5, the theoretical mean, by more than
twice the S.D., for the number ranges only between 1 and 9.

If we treat the 900 digits as 900 separate experiments with one
event each, instead of treating them as 36 experiments containing
25 events each, we have 1/10 as the chance for the appearance of
any particular digit, and hence the number of times any digit may
be expected to appear

=n2)±f-v/(%^g), approximately

= (900)tV± I V(900 X tV X TO )

-90±6.

The actual number of occurrences of each digit was as follows : —

Digit ....
No. of Occurrences

0
95

1
96

2
93

3

105

4
91

5
80

6

82

7
72

8
90

9
96

INTRODUCTION TO PROBABILITY AND SAMPLING 149

SO that the digit 7 showed the greatest divergence from 90 of any,
and this was only just three times the probable error.

[The Theory of Probability is older than that of Statistics. Todhunter, in
his History, states that ' writers on the subject have shown a justifiable pride
in connecting its true origin with the great name of Pascal.' The well-known
story of the latter being found, as a lad of twelve, tracing out on the hall floor
geometrical propositions which he had evolved in his own head is not. to be
wondered at, nor yet that at sixteen he wrote a small work on Conic Sections,
when one reflects upon the fame he was to win as a philosopher and writer,
as well as a mathematician, in his too brief life of thirty-nine years. He was
born in 1623 of a distinguished French family, and for the last half of his
life he suffered from the effects of a serious disease which contributed to turn
his attention from mathematics to religion and philosophy.

We learn from Todhunter how a certain gentleman of repute at the gaming
tables set Pascal pondering on a question of probability concerning the fair
division of stakes between two players who give up their game before its con-
clusion— an old problem cited in a work by Luca Pacioli as early as 1494. A
correspondence followed between him and Fermat, then probably the two most
distinguished mathematicians in Europe, and so began a science which has
fascinated at one time or another all great mathematicians from that day to
this.

The illustrious family of the BernouUis, friends of Leibnitz, who championed
his claim against that made by English mathematicians on behalf of Newton
to the invention of the Calculus ; De Moivre, an exile in England, owing to
the revocation of the Edict of Nantes ; Euler, Lagrange, and Laplace, who
worked out in algebraical form Newton's theory of gravitation for the motion
of the planets — all these had a share in building up the science of ProbabiUty,
often by investigating problems in games of chance, where the conditions can
be made mathematically perfect, so by careful analysis preparing the way for
the use later of the same principles in matters of greater importance.

It has been said that the development of the subject owes more to Laplace
(1749-1827) than to any other mathematician ; nor did he confine himself to
its theory : he would have earned fame by his astronomical applications alone.
His method was to take certain observations, and to determine by means of
probability whether the abnormalities present were merely the results of chance
or whether there was some as yet undiscovered but constantly acting cause
behind the phenomena observed. In this way he was led to highly interesting
and important results such as those relating to the theory of the tides, the
effect of the spheroidal shape of the earth on the motion of the moon, the
irregularities of Jupiter and Saturn, and the laws which govern the motion
of Jupiter's moons. It needs but a step in thought to pass from the dis-
cussion of such physical data to the statistics of social phenomena and the
causes which determine abnormalities met with in that field. Professor Edge-
worth, in making reference to books that have been WTitten on Probability at
the end of his excellent article under that heading in the Encyclopcedia
Britannica, remarks that ' aa a comprehensive and masterly treatment of
the subject as a whole, in its philosophical as well as mathematical character,
there is nothing similar or second to Laplace's Thiorie analyfique des
probabilifes.'J

CHAPTER XIII

GENERAL POPULATION.

SAMPLING [continued) — formula for probable errors

So far we have only considered the most simple case of random
samjjling when we take a sample of n independent events each of
which falls into one of two classes according to its nature, the
chance of entering either class being the same for every event :
we have dealt, that is to say, more particularly with non-measurable

characters. We pass on now to measur-
able characters which are distributed
among several classes according to their
size, so that a frequency distribution
table can be set up for each sample ; and
assuming that the population from which
the samples are drawn is homogeneous,
the samples themselves containing each
an adequate number of individuals, there
should not be greater differences between
one table and another than can be ac-
counted for by random sampling. It is
our object to discover how great such
differences may be.

Given a homogeneous population of N
individuals which we will suppose could
be distributed into a number of groups,
Yj individuals in the first group, Y2 in the
second group, Y3 in the third, and so
on, according to the size of the organ or
character under observation. Suppose a
random sample r)f n individuals be taken
from this population, and when they are
assigned to their several groups let the
frequency table now take the form shown,
with Pi individuals in the first group, y^
in the second, and so on. To find the probable error of y^, the
frequency observed in the kth group.

150

Class.

Frequency.

Yi
Y2

Y,

1st Group
2nd Group

A;th Group

N

sample.

Class.

Frequencj-.

1st Group
2nd Group

kth Group

2/2
Vk

n

SAMPLING — FORMULA FOR PROBABLE ERRORS 151

Consider the selection of the n individuals, one by one in succession,
to form the sample. When the first choice is made the probability
that we shall get an individual falling into the A;th group is, by defini-
tion, Y;fc/N, and the probability will remain practically the same for
each successive choice granted that N is considerable. We have thus
n independent events, the chance of success (falling into the A;th
group) for each being p(=Yfc/N) and the chance of failure being

nl =1— — -]. The case is therefore analogous to the one pre-

viously considered to which the binomial distribution is applic-
able, so that the frequency to be expected in the ^th group is np
with S.D., Oy =Vnpq ; i.e. yk=np with a p.e.=0-6745Vwi>g'-

Now in practice the numbers Yj, Yg, Y3 . . . would not be known,
and hence the true value of p would also be unknown, but since
yk=np, approximately, when the sample is of adequate size, we
shall get a fair idea of the probable error involved by taking
p—yjn, where y^ is the actual frequency observed in the A;th group.

. (1)

Hence, CT^,^=wpg=i/fc(l— ^)=yk(^l— -

and the frequency in the kth group

=y,±0-6745^

(2)

The size of the S.D. is under ordinary conditions a test of the
adequacy of the sample, for the frequency in the kth group, if due
simply to random sampling,
should not differ from its
expected value by more than
3cr„ and ct„ should therefore

be small compared with y^.
itself.

To find the correlation between
the frequencies in any two
groups of a sample distribution.

Let the expected frequencies
in the various groups of the
sample be denoted by y^, y^,
. . ., yj^, . . ., and suppose an
error Sy^ in y^. is associated
with errors Sy-^, Sy^, . . . , Sy„ ... in y^, y^,
require then the correlation between yj^ and y^

Class.

Expected
Frequency.

Observed
Frequency.

1st Group
2nd Group

kth Group
sth Group

2/2

y>

2/2 + ^2/2

yk+^yk
y.+^y.

n

We

152

STATISTICS

Now although the group frequencies may change relative to one
another, the total sum of frequencies in all groups is not affected,
because the n individuals of the sample make up its composition in
each case : to keep n constant the group frequencies must adjust
themselves accordingly, which explains the correlation between
them. Hence to compensate for an excess, Syj^ (assuming §2/^+'^'^),
of frequency in any one group there must be a defect (— St/^^) shared
among the other groups, and the fairest way of sharing will be in
proportion to the expected frequencies in the several groups.

But the total frequency divided between groups other than the
kth is (n—yj^), so that the proportion of (— S^/fc) due to the sth group
is yj{n-yj,), thus

Vs

S2/,= -^^(-8y.).

Therefore,

S^/fc • 8^3

8y\

n

2/. 1-

Vk

•Vic

n ct2

Vk

. (3)

by (1).

FIRST SAMPLE.

Size of Organ

or Character

observed.

Frequency of
Observations.

First Moment.

Second Moment.

^1

3/1

y-i

Vk

^kVk

^\yi ,.
x\yk

n

2(X2/)

^x^y)

This gives the product moment of the deviations from yj^ and y,
in one particular sample ; summing for all such samples, remem-
bering that by definition the coefficient of correlation between ?/^

SAMPLING — FORMULA FOR PROBABLE ERRORS 153

and 2/s is ^y y ^^i^Vk • ^ys)h'^v ^v > where v is the total number
of samples, also a'^y =I!Sy^,./i>, we have

vr

cr„ CT„ =•

Vuv^ Vi. y

V -Vk-

Therefore,

1 Yi^y,

. (4)

gives the correlation required.

To find the p.e. of the mean of a sample of n observations. Let a
frequency table be drawn up in the usual manner showing the
number of observations 2/i> 2/2 • • • corresponding to organs of
different sizes x^, x^ . . .

The mean referred to some fixed point as origin is then given by

also the mean square deviation of the sample referred to the same
fixed point is i^-^, say, given by

and H-\—W-^a'^

where a is the S.D. of the sample.

For another sample of the same size the frequency distribution

SECOND SAMPLE.

Size of Organ

or Character

observed.

Frequency of
Observations.

First Moment.

Xk

y-i + ^v-i

Vk+^Vk

^iiVl + %2)

^k{yk + ^Vk)

n

My+^y)

may be slightly different, say, Vi+hij^, 2/2+^2/2' • • •> ^^i^ conse-
quently the mean will also be different, say,

M+8M=[a:i(t/i+82/i)+X2(2/2+8t/2)+ . . . ]/n.

154 STATISTICS

■^and, by subtraction,

8M=(^i8yi+a;28t/2+ ■ • •)/» • • • (5)

Now we want to determine the S.D. of the different values of M
found among the different samples, and that is given by

where 2J denotes summation for all samples and v is the number of
sainples. This suggests that we should square both sides of
equation (5), getting

n^ . m^ =- x\Sy\-\- . . . -\-2x^XoSyiSy2+ • . .

TheTeioTe,n^ .va\^x\va^yi+ . . . -\-2xixJ-'^.-v)-\- . . .,
by (3). Hence, n-aking use also of (1),

■ yi\ , 2^i?/i . x^y^

nV2M=^Wl-|) +

-{x\yx+ . . . )-hAy\+ . . . +2x,y, . x^y,-{- . . .)
n

=nf^\—{Xiy^-\- . . .)^
n

=n{H-%-W).
Thus a\={f^\-M^)/n=a^/n,
and the probable error of the mean =0 •6745a/ Vw . . • (6)

The p.e. in the arithmetic mean found by taking a random sample
of n events is a measure, so to speak, of the failure to hit the absolute
mean, and it follows that the precision of the sample, the accuracy
of aim at the mean, would be not unfairly measured by some
quantity proportional to the reciprocal of the above expression,
namely, \/w/0-6745o-. With such a measure the precision would
evidently be increased if the number of observations in the sample
were increased, being proportional to the square root of their
number.

[It is desirable to draw a distinction here between what have been
termed biassed errors and unbiassed errors ; errors due to random
sampling are of the second class for there is, by hjrpothesis, no

[* We do not know the true mean for the population as a whole, but we take
in place of it M, the value given by the sample, which we may do with little
error if n is large. Similarly (t is the S.D. of the sample.]

SAMPLING — FORMULA FOR PROBABLE ERRORS 155

reason Avhy they should be in one direction rather than in another.
Biassed errors, however, all tend to be in the same direction and
they may arise in different ways, e.g. they may be due to faults of
omission or commission on the part of the observer himself : he
observes either carelessly or badly, omitting certain factors which
ought to be taken into account, or so measuring or classifying his
results that they appear always larger or less than they really are
in fact.

Sometimes, although the bias is known to exist, it may be im-
possible to correct it : the most one can do is to bear it in mind
and allow for it in using the results. A familiar example of this
occurs in the collection of household budgets from the poor to find
their standard of living, where it is only possible to get particulars
from the more intelligent and thrifty class among them.

Whereas in the case of unbiassed errors due to random sampling
we can diminish the probable error of the average by increasing
the number of observations, the same is not true of errors which
are biassed, for suppose an error e in excess be made in each of
n observations x^, x^, . . . a:„, the effect upon the average is to
increase it from

Xi+x^+ . . . +a:„ (a:i+e)+(a:2+e)-f . • . +(^„+e)

tQ ,

n n

i.e. from

•^11 •''2 1" • • • -rX„ Xi-\-X2-\- . . . -\-Xji

to -f-e,

n n

so that the average is over-estimated by precisely the same amount.
If, therefore, we know that bias exists, it is well, if possible, to
correct it in each observation, for by so doing we change biassed
into unbiassed errors, and though our corrections may be somewhat
wide of the mark, the resultant error will then be diminished by
increasing the number of observations : e.g. a farmer offers 400
sheep for sale and, being anxious to make a good bargain, he asks
a higher figure for them than he is in reality prepared to take ;
let us suppose that this excess is 2s. 6d. for each sheep, then clearly
the average price per sheep at which he is prepared to sell will be
less than the amount he asks by 2s. 6d. also. But now suppose the
buyer, a simple person knowing little of the prices of sheep and
less of the ways ^ of men, goes through the flock one by one and
makes the error of offering a price either much above or much below
what the seller is prepared to take ; even if his unbiassed offers

156 STATISTICS

difiFer by as much as 10s. for each sheep from the seller's reserve
price, so long as they are random in direction, i.e. sometimes too
much and sometimes too little, the resultant difference in the
average from what the seller is prepared to take will probably not
greatly exceed §10s./V400, or 4d. per sheep.

We can sometimes diminish the effect of bias, even when its
extent is unkno\^Ti, by working with the ratios of the quantities
affected instead of with the quantities themselves : e.g. suppose
biassed errors, ei and €.2, enter into the measurement of the variables
Xj^ and X2, both in excess, the ratio of the variables then

= K+ei)/(a;2+e2)

=^(i+l^)(i+:

=— ( l + -^)( 1— — +higher powers of €0)

-1

=5l+f5.

^2

if we omit higher powers of ej and eg than the first on the under-
standing that they are both comparatively small. Suppose, for
example, there was an error of 5 per cent, made in measuring x-^
and an error of 3 per cent, of like sign in measuring x^ then the
resulting error in x.^\x^ would be 5 per cent. — 3 per cent. =2 per cent.
Clearly the same holds good also if the errors are both in defect.
This explains why a comparison of results arranged, say, on the
index number principle may be trustworthy, although the method
of formation of the numbers themselves may be in some respects
faulty, granted that the same faults are repeated each year so as
to produce like errors, i.e. the bias is to be unchanged in character.
To correct the faults in one case and not in the other would prejudice
the success of the method, since it depends upon the errors counter-
acting one another.]

Example (1). — To illustrate the important result we have obtained
for the p.e. of the mean of n observations let us return to the experi-
ment of selecting 900 random digits. The distribution actually
obtained, and the theoretical distribution to be expected in the

SAMPLING — FORMULA FOR PROBABLE ERRORS 157

long run if the experiment were repeated several hundred times and
the average taken, are shown in the following table : —

Table (37). Distribution of 909 Random Digits.

Digit.

Frequency
Observed.

Theoretical
Frequency.

Digit.

Frequency
Observed.

Theoretical
Frequency.

0
1
2
3
4

95
96
93
105
91

90
90
90
90
90

5
6

7
8
9

80
82
72
90
96

90
90
90
90
90

It is a simple matter to calculate the mean and S.D. for the dis-
tribution from this table in the usual way ; the results are : —

Observed mean=4-38 ; S.D.-2-911
Theoretical mean=4-50 ; S.D.=2-872.

Thus the p.e. of the mean based on the sample

= ±0-6745 X 2-911/^/900
= ±0-065,

and 4-38 differs from 4-50 by less than three times the p.e.

The 36 averages of samples of 25 events apiece were also calcu-
lated, and the following were the results obtained : —

2-76, 3-32, 3-68, 3-72, 3-72, 3-72, 3-76, 3-80, 3-92, 3-92, 408, 4-12,
4-16, 4-16, 4-16, 4-28, 4-36, 4-40, 4-40, 4-40, 4-44, 4-60, 4-64, 4-68,
4-72, 4-72, 4-76, 4-88, 4-96, 5-00, 5-00, 5-00, 5-08, 5-28, 5-40, 5-72.

The mean of this distribution= 157-72/36=4-381, and the
S.D. =0-612. But the S.D. of the whole distribution of 900 digits
=2-911, and therefore the S.D. of the distribution of averages of
samples of 25 digits should be 2-911/'v/25=0-582, differing from
0-612 by about 5 per cent.

To find the p.e. of the sum or difference of two variables. Let the
mean values of the two variables be denoted by y and z, so that
deviations from these values found in a particular sample may be
denoted by hy and hz. If then we wTite

u=y-\-z

we have

8m=S?/+8z

(7)

158 STATISTICS

To find the S.D. of u we therefore require S{hu^)jv, where the
Z denotes summation for all samples and v is the number of samples.
But, squaring both sides of equation (7), we have

8w2=§y2_|_§22_|.282/8z.

Thus Zhu^=Ehy'^+Shz^+22:{hyhz),

where the summation extends to all samples. Hence

iCT\= va'^y+ va\-\-2vay(T^ry^

or a\=a%-^a\-\-2Tj,a^a,

where r„. is the correlation between the variables. And the
p.e.=0-6745cT„.

The p.e. of the difference of two variables follows at once by
changing the sign of z throughout ; for, if

v=y-z,

we have Sv^=Sy^~\-hz^—2SySz,

and a\=a%-\-a\—2T^^aya^.

Generally, if Xj^, x^, . . . a;„ be the mean values of n variables,
and if hx^, hx^, ■ . ■ Sx^ denote deviations from these values in
a particular sample, we may write

U = Xj^ + X2-{- . ,

. . -\-x„

and

8w=8.ri+8.r2+

. . . +8a;„.

Thus

2'8w2=i;8.ri2+ . .

. +2Z{dx,8x,)+

whence

<=o\+ ■ • .

+2r, , o-^ (7, + .

Important Corollary. If y and z are quite independent so that
Ty^ is zero, the p.e. of their sum and the p.e. of their difference
have the same value, namely, the square root of the sum of the
squares of the p.e.'s of y and z themselves, which

=0-6745v/(a\+a\) . . . (8)

This result is exceedingly important, because it can be directly
used to test whether a difference between two samples is accidental,
i.e. whether it is such as might arise through sampling, or whether
it implies a real difference between the two populations from which
the samples are selected. An example will illustrate the pro-
cedure : —

Example (2). In a study of Minimum Rates in the Tailoring
Indtistry, by R. H. Tawney, a table is given (p. 114) which suggests

SAMPLING — FORMULA FOR PROBABLE ERRORS 159

that ' in the north of England women work in the tailoring trade
when they are young ... in London and Colchester they have
to work when they are older.' Taking some figures from that
table we find : —

District.

Workers over
36 years old.

Workers at
all ages.

Proportion
over 35.

London and Essex
Manchester and Leeds .

11,718
4,029

35,316

21,822

0332
0185

The difference between the proportions over 35 years of age
= (0-332-0185) = 0-147.

Let us suppose for the moment that this difference is not significant
of any real difference in conditions between the two districts, but
18 merely due to random sampling. In that fcase the most natural
value to assign to the true proportion of women workers over 35
for the trade as a whole, as given by these figures, would be

11,718+4,029 _15,747_,.g^^
35,316+21,822 57,138

The S.D. for the first sample (London and Essex) would then be

a^== ^/{pq/n)^ ^[0-216 X 0-724/35,316],

and for the second sample (Manchester and Leeds) would be

(72= VL0-276X 0-724/21,822].

Hence the p.e. for the difference between the proportions in the
two samples would be roughly

= tV'(<7'i+^'2)> by (8),

= |V'[0-276 X 0-724(1/35,316+ 1/21,822)]

= 1 V'[0-276 X 0-724/13500]

==00026.

The actual difference between the proportions, 0-147, being much
more than 3(0-0026), is certainly significant of a greater difference
between the two populations than can be explained by random
sampling alone.

160 STATISTICS

Another method of attack would be to assume a real difference
between the two populations, if other considerations led us to
suspect such a difference, and to find whether such a difference could
be disguised by random sampling. In that case the proper pro-
portion to assume for the first sample would be 0-332, giving

C7j= V[0-332 X 0-668/35,316]= ^628/10*,

and for the second sample the proportion would be 0-185, giving

(j2= V[0-185 X 0-815/21,822]= ^691/10^.

Hence the p.e. for the difference between these two proportions
due to random sampling would be

= WK'+^2'), by (8),
= fI^V(628+691)

=0-0024.

The actual difference is 0-147, which certainly could not be out-
balanced by an error in the opposite direction due to random
sampling, because it is much more than three times the probable
error due to sampling.

Sometimes we have to test the difference, not between two
simple proportions, but between two sample distributions. In
that case the mean of each sample may be calculated so that the
difference (M^— Mg) between the means is known ; to find out
whether or not it is significant of some real difference between the
two populations from which the samples are drawn, (M^— Mg)
is compared with its p.e., namely

0-6745a/Kmi+ct\2),
or 0-6745 V((T2i/Wi+(T2,,/r?.2) . . . (9)

where n^ and %2 are the numbers of observations in the two samples
respectively, and a^, 02 are the S.D.'s of the samples. Unless
(Mj— M2) is definitely greater than some two or three times this
expression we cannot be verj' sure that the difference between M^
and Mj may not have arisen merely through random sampling,
and it may quite likely not be significant * of any real difference
between the two populations as regards the organ or character
which is under consideration.

[* It should be observed that the S.D. provides a wider margin for significance
than the p.e., because a range of approximately 3 p.e. =3'§ff = 2<r only. It is
quite safe therefore to attach no great significance to a difference which does
not exceed two or three times the p.e.]

SAMPLING — FORMULA FOR PROBABLE ERRORS 161

Example (3). — Statistics have been collected to test whether there
is any significant difference between the eggs laid in general by
cuckoos and those laid by them in the nests of particular species
of foster parents. Results of the following kind were obtained
[see Biometrika, vol. iv., pp. 363-373, The Egg of Cuckidus Canorus
(2nd Memoir), by 0. H. Latter] : —

Number

Mean

S.D.

(mms.)

Signi-

Group.

of

Length

ficance

Remarks.

Eggs.

(mms.)

Test.

Eggs of the Cuckoo

race in general

1572

22-3

0-9642

. .

Eggs laid in nests of —

Garden Warbler .

91

21-9

0-7860

70

Significant.

White Wagtail

115

22-1

0-7606

1-6

Not significant.

Hedge Sparrow

58

22-6

0-8759

3-75

Probably significant.

The difference between the mean lengths of eggs laid in the nests
of garden warblers and those laid by cuckoos in general

= 22-3— 21-9=0-4 mms.

The p.e. of this difference

= 0-6745V[(0-7860)'^/91 + (0-9642)2/1572], by (9),

=0-6745V(0-007380)

=0-058.

Hence the significance test
=0-4/0-058=7-0,

and we conclude that the difference in length between the two
classes of eggs is certainly significant. Similarly the other cases
may be tested.

In the example just given, to find out whether one population
differed from another, the arithmetic means have ]>een compared ;
but the mean alone will scarcely serve to establish the identit}'' of
any population. For example, we can conceive of two distinct
races of men, both of the same mean height, but one race embracing
a number of giants and dwarfs. Of course if we agreed to define
two races as identical when they have the same mean heights, there
would be nothing more to be said, but that would certainly only
be a very rough-and-ready attempt at classification.

Taking into consideration only the character of height, a further
step in definition would be to measure the mode or most fashionable

h

162 STATISTICS

height, and the dispersion or variability — absolute : the standard
deviation, and relative : the coefficient of variation— of the two
races. Then, after comparing heights with sufficient detail, the
attention could be turned to innumerable other characters, skull
and body measurements, physical, mental, and even moral
attributes.

Clearly the difficulty of definition and of estabhshment of identity
grows as we pass along the scale from physical to moral. Moreover,
other statistical constants must be requisitioned when the question
of the existence and degree of relationship between two organs or
characters is to be determined. As the S.D. and the C. of V. serve
to measure the amount of variability, so the coefficient of correlation
comes in to measure the amount of likeness or association. Further,
and especially in problems of inheritance, the coefficient of regres-
sion must be measured. It might seem at first sight hopeless to
try and measure the correlation between two such characters as
athletic capacity and health in the same boy, or between the
truthfulness of one boy and that of his brother ; but the genius of
Karl Pearson has gone some way to solve even this difficult problem
by means of a system of adjectival instead of numerical classifica-
tion [see Phil. Trans., vol. 195a, pp. 1-47, On the Correlation of
Characters not Quantitatively Measurable, and, as an exceptionally
interesting application of the method, see Pearson, On the Laws of
Inheritance in Man, ii. ; On the Inheritance of the Mental and Moral
Characters in Man and its Comparison with the Inheritance of the
Physical Characters; Biometrika, vol. iii. pp. 131-190]. In short,
for a full and exact definition of a population of any kind, human
or otherwise, it is necessary to measure not only the means, but all
the more important statistical constants, modes, medians, S.D.'s,
C.'s of v., coefficients of correlation and regression, and so on, and
it is no less necessary to calculate also their probable errors if we
are to test the real significance of such differences as are observed
in these constants between two samples from the same or from
different pojoulations.

The probable errors for the more important constants, some of
which are only introduced later in the book, are collected together
in Table (38) for reference. The proofs in general are a little intricate
and would be lacking in interest to the ordinary person, who is
satisfied to take algebraical analysis on trust so long as he under-
stands the nature of the results he uses, but the more mathematical
reader who is anxious to see proofs may refer for some of them to
Biometrika, vol. ii,, pp. 273-281, Editorial, On the Probable Errors

SAMPLING — FORMULA FOR PROBABLE ERRORS 163

oj Frequency Constants, which has been freely consulted on the
subject here.

The usual notation is adopted, n being the total number of
observations in the given distribution, supposed normal in general,
a the S.D., etc.

Table (38). Probable Error.s of Statistical Constants.

statistical Constant.

Proba
0-674

ble Error (=0G745S.D.).

Any observed group frequency, y

5x V[y(l-2/A0]

The mean of a distribution of any type

„

<t/V71

The S.D. of a normal distribution, a .

„

alV-ln

[The second moment about the mean, fio

„

(T-V2/11

■ „ third „ „ „ ^3 •
[ „ fourth „ „ „ f^i .

"

0-V96/W

The coefficient of variation, v ■ .

>»

V2v\- MOO/ J

The coefficient of correlation, r

»

(l-r^)/Vn

The correlation ratio, rj . . . .

„

{l — if)/Vn, nearly

fx, as determined from (X-X) = ?-'^'(Y- Y),

V

when Y is given .....

,

cr^Va-r')

Y, as determined from (Y-Y) = r^^(X-X),

^ when X is given .....

Distance between mode and mean in a skew
distribution ......

Skewness ......

/3., (which should = 3 for a normal distribution)
^i( „ „ =0 „ „ )

^K

J5

fr.Vd-r^)

(rV(3/2u)

V(3/2n)

0

V(6/n)

Example (4). — In the example which follows are given data
necessary for testing the significance of differences in variability
as well as in mean values. They represent an attempt made to
find whether members of a particular species of crab caught in
shallow water differed with regard to certain characteristics from
those caught in comparatively deep water [see Biometrika, vol. ii.,
pp. 191 et seq., Variation in Eupagurus Prideauzi, by E. H. J.
Schuster]. Only a few of the results are recorded here, to two
decimal places ; the reader will find it a valuable exercise to verify
for himself the p.e.'s given in each case.

164

STATISTICS

Measurement Made.

Sex.

Locality.

Mean (mm.).

S.D. (mm.).

-

C. of V.
per cent.

Carapace length

Male
Female

Deep water
Shallow ,,
Deep ,,
Shallow ,,

8-59 ±0-05
841 ±0-04
7-54 ±0-03
7-12db0'02

l-67±0-04
1-19±003
0-94 ±0-02
0-8G±0-02

L945

l249|o.28
l2-12±0-25

Difference of Means (mm.).

Difference of S.D.'s (mm.).

Difference of C.'s of V.
per cent.

Sex.

0-18±0-07(poss. sig.)
042i:0-04(sig.)

0-18d-0-05(prob.sig.)
0-08±0-03(poss. sig.)

l-70db0-58(poss. sig.)
0-37±0-37 (not sig. )

Male
Female

The significance or otherwise of differences between variabihties
in the case of cuckoos' eggs (p. 161) might be tested in the same way.

CHAPTER XIV

FURTHER APPLICATIONS OF SAMPLESTG FORMULA

We have been discussing in the last chapter how to test two samples,
supposed each to contain homogeneous material, to find oiit whether
they belong to the same or to different types of population, but
the further question often arises as to whether a sample is or is not
homogeneous.

Example (1). — To this we may obtain a partial answer by working
out the statistical constants of the sample and their p.e.'s in order
to compare them with the corresponding constants for a sample or
series of samples believed to be homogeneous and of the same
type. For example, Professor Karl Pearson has measured the
skulls of skeletons of the Naqada race, excavated in Upper Egypt
by Professor Flinders Petrie and presumed to be some 8000 years
old, and he places his results for comparison alongside those
for certain other races admittedly homogeneous [see Biometrika,
vol. ii., p. 345, Homogeneity and Heterogeneity in Collections of
Crania'] : —

c, _...-.„

Number of
Observations.

Variability (mm.).

Skull Length.

Skull Breadth.

Skulls ^

Living
head.s

'Ainos

Bavarians .

Parisians

Naqadas
,EngUsh

Cambridge undergrad'tes

English criminals
^Oraons of Chota Nagpur

76

100

77

139

136

1000

3000

100

5-936
6-088
5-942
5-722
6-085
6-161
6046
5-916

3-897
5-849
5-214
4-612
4-976
5-055
5014
4-397

Mean Variability

5-987

4-877

1U5

166 STATISTICS

The S.D. of the variability of skull length calculated from this
series=0-129 mm. and of the variability of skull breadth=0-545 mm.,
and these supply standards for valuing the differences between the
Naqada and the mean variabilities.

Another method of procedure is to take a random sample out of
the sample itself, assuming the latter is large enough to admit of
an adequate sub-sample, and to compare the constants of the
whole and part. When they do not dififer beyond the limits allowed
by random sampling the inference is that the whole may be treated
as a homogeneous class if judged by this test alone.

Example (2). — In an interesting and important memoir, On
Criminal Anthropometry and the IdentificaHon of Criminals, by W. R.
Macdonell [Biometrika, vol. i., pp. 177 et seq.], the author uses this
method to test the homogeneity of a class of 3000 criminals by
measuring also a random sample of 1306 criminals out of the 3000.
He obtained, for example,

S.D. of head length--6-04593i0-05265 mm., for the 3000 criminals ;
= 6-00247 ±007922 „ „ 1306

The difference between the variabilities in the sample and sub-
sample, by result (8) on p. 158,

^0-04346±V[(0-05265)2+(0-07922)2]
=0-04346±009512

which is certainly not significant. If the same holds good with
regard to the means and other constants, then the whole may be
said to be homogeneous so far as this test goes.

Example (3). — Another example may be given from the memoir
on Variation and Correlation in Brain Weight, by Raymond Pearl,
[Biometrika, vol. iv., pp. 13 et seq.]. The author wished particularly
to investigate the change of brain weight with age ; on the hypo-
thesis that the weight of the brain reaches a maximum between
the ages of 15 and 20, remains unchanged from 20 to 50, and then
begins to decline and so continues till death, the material was
divided into a ' Young ' series, ages 20 to 50, and a ' Total ' series
including all between 20 and 80. The ' Young ' series thus formed
a selection from the ' Total ' series, but a selection based on age
and not on brain weight. If there were no con-elation between
age and brain weight, this selection, based as it is on age, would,
of course, be random as regards brain weight. Now correlation
does exist between the two, but it is so slight that, within the limits

FURTHER APPLICATIONS OF SAMPLING FORMULA 167

of error, the ' Young ' series does form practically a random sample
of the ■ Total ' series, as is shoAMi by the following figures : —

Difference in Variation Constants between Young and
Total Series (written with a positive sign ^VHEN the
Young Series gives the greater value).

Swedes
Bavarians

Male.

Female.

S.D. I C. of V. i S.D. I C. of V.

+ 2-851 ±4066 +0122±0-291 + 4-786 ±o-46o +0-271 i 0-435
-1-888 + 3-5561 -0-173+0-234 l-10-357+3-909|-0-941+0-320

Thus in only one case, that of the Bavarian females, is the differ-
ence between the variabilities, S.D. or C. of V., of the two series as
gi-eat as its probable error, and even in that case the differences.
10-357 and 0-941, are not three times as large as their respective
p.e.'s, 3-909 and 0-320. Dr. Pearl concludes from these and similar
results that ' the series are reasonably homogeneous in other respects
than age.'

The reader is recommended to test Ms knowledge of the formulae
for probable errors bj^ applying them to the following examples.
Dr. Alice Lee, in a note on Dr. Ludwig on Variation and Correlation
in Plants [Biometrika, vol. i., p. 316] makes use of the statistics
relating to Ficaria Verna in Example (4). Those in Example (5)
are taken from among a large number of others in the highly
interesting memoir, On the Laws of Inheritance in Man, by Professor
Karl Pearson and Dr. Alice Lee [Biometrika, vol. ii., pp. 357 et seq.]
cited once before.

Example (4). — Variation and Correlation in Ficaria Verna.

No. of Observations.

Mean No. of
Petals; S.D.

Mean No. of
Sepals; S.D.

Correlation between

No. of Sepals and

No. of Petals.

1000 (Greiz A)
1000 (Greiz G)

8-286; 1-3382
8-232; 0-9954

3-695; 0-8524
3-437; 0-7033

0-2439+0-0201
0-2480+0-0200

We have here all the data necessary to find the p.e.'s of the
means, variabilities, and correlations, and we wish to know whether

168

STATISTICS

the differences between the means and variabihties of the A and G
plants can be accounted for by random samphng alone.
For examjDle, the difference between the petal means

= (8-286- 8-232) ± I
=0-054±0-035.

; 1 -3382)2 (0-9954)2"|
1000~ 1000~J

Clearly this difference, being not so great as twice its p.e., is not
significant and may quite well be due to random sampling.
Again, the difference between the petal variabilities

(1-3382)2 , (0-9954)2"|

2000

2000

= (l-3382-0-9954)±|

=0-3428±0-025

which is certainly much too great to be explained away by random
sampling merely.

Similarly the differences betAveen the sepal means, between the
sepal variabilities, and between the correlations, may be tested for
significance by comparison with their p.e.'s.

Example (5). — Size and Variability of Stature in the
Two Generations.

Father.

Mother.

Son.

Daughter.

Mean height (in.)

S.D. (in.) .

C. of V. (per cent.)

67-68 ±0-06
2-70±0-04
3-99±0-06

62-48 ±0-05
2-39 ±004
3-83±006

68-65±005
2-71 ±0-04
3-95±0-06

63-87 ±0-05
2-61 ±0-03
4-09 ±005

The student in this case might use one of the formulae for the
p.e.'s to find the number of fathers, mothers, sons, or daughters
observed when the p.e.'s are known, and then the remaining p.e.'s
might be verified v/hen the numbers of observations are found.

As evidence of ' assortative mating,' the tendency of like to
mate with like, the following particulars are given, based on 1000
to 1050 cases of husband and wife : —

Correlation between stature of husVjand and stature of wife = 0-2804±0-0189

span ,, ,, ,, span ,, ,, =0-1989±0-0204

,, forearm ,, „ ,, forearm ,, ,, =0-1977±0-0205

To measure the average intensity of inheritance, the extent of

FURTHER APPLICATIONS OF SAMPLING FORMULA 169

resemblance between parents and children in any character, co-
efficients of correlation are calculated such as the following : —

Coefficient of Correlation

between stature of father and stature of son =0-514±0-015

,, ,, ,, ,, ,. daughter = 0'510±0-016

,, mother ,, ,, „ son =0494±0-016

„ ,, ,, „ ,, daughter = 0-507±0-016

[In verifying the p.e.'s for this case take the number of observa-
tions to be 1024.]

One more extract may be quoted, a prediction table, giving the
probable mean stature of sons of fathers of given stature, and
so on : —

Son's probable stature = 33"73 + 0"516 (father's stature) ± 1 "56

Daughter's „ „ = 30-50 + 0-493 ( „ „ )±1-51

Son's „ „ := 33-65 + 0-561) (mother'sstature)± 1-69

Dauf^hter's „ „ = 29-28 + 0-554 ( „ „ )±l-52.

All values given in this examjDle for the la.e.'s should be
verified.

Before we consider further applications of these principles to
questions of a somewhat different kind, let us imagine a very
simple though artificial illustration. Suppose we have 999 sheep,
each one ticketed, the numbers on the tickets running from 1 to
999. Also suppose 666 of these sheep are white and 333 are black,
so that, if we pick out any one at random, the chance of it being
black is 333/999 or 1/3. Let us call picking a black sheep a 'success,'
then p- 1/3, ^=2/3.

We proceed now to select 99 sheep in succession at random
from the flock with the understanding that each sheep is returned
into the flock before the next is j)icked out. This insures that
the chance of a success at each selection remains equal to 1/3 and,
of course, there is nothing to prevent the same sheep being picked
more than once. The selection might practically be made by
placing in a box 999 tickets, numbered from 1 to 999, one to corre-
spond to each sheep, then picking out 99 of them in succession,
being careful to replace each and to shake up the box before j)icking
out the next ; if there were absolutely no difference between the
tickets, such as would cause one to be picked more easily than
another, the selection made in this way would be random in the

170 STATISTICS

sense required, and the tickets so chosen would determine wliich
sheep were to be taken and which left.

The proportion of black sheep to be exi)ected in such a random
selection of 99 is 1/3, but, if we only perform the experiment once,
it is quite likely that the proportion we actually get will differ from
1/3 by an amount

= 0-6745 V(^g/w)

=0-6745V(J . I • A)

= 1/31, about,

while it is unlikely that the proportion will differ from 1/3 by much
more than 3/31, or 1/10.

Conversely — and it is rcall}- the converse which is useful in prac-
tice— if we do not Icnow the proportion of black sheep m the whole
flock, we may get a fair estimate of it by taking a random sample
of 99 sheep (anj' other number will serve the purpose, but the
larger the better for accurac}'), and if we find that in this sample
there are 33 black sheep, i.e. 2>=33/99=l/3, it will appear that
the value of x> for the whole flock is 1/3, subject to a probable error
0 -Ql 4:5 \/{2)qj'n) in excess or defect, i.e. the true proportion for the
whole flock may quite likely differ from 1/3 by as much as 1/31,
but it is unlikely to differ by much more than 1/10. It should be
noticed that the calculation of the probable error in this converse
case is based upon the value of p given by the sample taken, for
that is the only value of which we have knowledge.

Too much stress can scarcely be laid on the fact that the samples
chosen must be absolutely unbiassed, otherwise the use of the
formula) np and \/{npq), or the corresponding proportional formulae,
cannot be justified : each sheep in our illustration must have the
same chance of being picked, and no one selection is to have any
influence on another. The failure to appreciate this essential
point has led to no little waste of time and effort in the collection
of valueless statistics.

The method of sampling has been employed in a way at once
interesting and useful by Dr. A. L. Bowley, and, as some of this
work has barely received the attention it deserves, it may be well
to explain two of his experiments in some detail.

The first was of interest because its results could be tested by
an examination of the original record from which the sample was
taken. The details concerning it are abstracted from the Journal
of the Royal Statistical Society, September 1906.

Example (6). — Bowley sampled the dividends paid by 3878

FURTHER APPLICATIONS OF SAMPLING FORMULA 171

companies as quoted in the Investors' Record. His sample con-
sisted of 400 of these companies, i.e. about 10 per cent., selected in
a purely arbitrary fashion thus : the investigator took a Nautical
Almanac and noted down the last digits of one of the tables, record-
ing them in groups of four, but if any particular group gave a
number bigger than 3878 he rejected it. In this way each of the
numbers between 1 and 3878 had an equal chance of selection (for
numbers under four figures would appear like 0327, 0042, 0009,
which would be taken to represent 327, 42, 9 respectively), and the
selection of one had no influence on that of any other. The com-
panies in the Investors' Record were numbered consecutively, and
the dividends corresponding to the 400 arbitrary numbers obtained
formed the sample with which Bowley worked.

After making some interesting deductions with regard to the
average for the whole distribution, to which we shall return pre-
sently, he proceeded to forecast the grouping of the original com-
panies as to their dividends hy setting out the grouping discovered
in the sample 400, as follows, using the standard deviation in place
of the probable error as the error due to random sampling ; —

Table (39). JJistribution of Dividends paid by a
Sample of 400 Companies.

(1)

(2)

(3)

(4)

Dividend.

Sample of

400
Companies.

Percentage of Sample
Companies in each Class.

Percentage of
all Companies
in each Class.

Nil

£1 to £2, 19s. 9d.
£3 to £3, 9s. 9d.
£3, 10s. to £3, 19s. 9d.
£4 to £4, 9s. 9d.
£4, 10s. to £4, 19s, 9d.
£5 to £5, 19s. 9d.
£6 to £7, 19s. 9d.
£8 to £10, 19s. 9d.
Above £11

28
6
37
71
64
53
60
48
29
4

7 with S.D. = l-27

U

9| .. -1-46
17| „ -1-90

16 „ =-1-83
13i „ =1-68

15 ,. -1-78

12 ., -1-63

7i „ -1-29

1

6
1-5

8-4
18-8
17-3
13-8
17-7
10-8
38
1-9

In col. (3) the S.D. for each group was calculated as follows : —
for the first group : out of 400 possible events we have 28 successful
events, meaning by ' successful ' here ' a company paying no
dividend,' thus

^-: 28/400, g-= 372/400.

172 STATISTICS

Hence the S.D. of the frequency in the fii'st group

=V[28(i-AV)]

= V(28x372)/20
=5-1.

Since this is for a sample of 400, the S.D. of the percentage* frequency
in the first group

-i(5-l)-l-27.

The other S.D.'s are calcuhxted in the same way, but when the
number in a class is very small the forecast can scarcely be relied
upon and consequently the S.D. is not inserted.

It will be noted, by comparing with the numbers in col. (4),
showing the corresponding percentages for all the 3878 companies,
that every forecast was remarkably good except one, class £8 to
£10, 19s. 9d., where the error approaches three times the S.D., and
the exception Avill serve as a warning that, in working with samples,
the unexpected sometimes happens. Professor Edge worth, in his
Presidential Address to the Royal Statistical Society (1912), points
out that the method appears to be a permanent institution in
the Statistical Bureau at Christiania, where it has given very good
results. These can be checked or ' controlled ' for safety if complete
statistics are obtainable under some heads. He faii'ly sums up the
utility of sampling when he says that ' we may obtain from samples
a general outline of the facts — often sufficient for the initiation of
a project like that of insurance— rather than the features in detail.'

Bowley also divided up his 400 random samples into 40 groups
of 10 companies each, and calculated the average for each group.
The S.D. for these 40 averages was found in the usual way, giving
0-775. But since this wsiS the S.D. for averages of 10, we conclude
that

(the S.D. for the distribution of the400companies)/'\/10=0-775
i.e. the S.D. for the distribution of the 400 companies=0-775y'10.

Hence, applying the same principle again,

the S.D. of the average of the 400 sample companies

-=0-775V10/\/400
=£0-122.

[* It would not be correct to take v'[7(l - ti^ij)] as the S.D. of the percentage
frequency in the first group ; this value would be double the true value, namely,
J v'[28(l - i\%)] = h \^n(^ ~iIt))], because the accuracy is increased by increasing
the number of events in a sample, and the sample here is really 400 and not 100.]

FURTHER APriJCATIOXS OF SAMPLING FORMULA 173

Now the average of the 400 samples turned out to be £4-7435.
Hence it was judged that, if this was a fair selection (and the rando)u
method adopted was such as to make it fair in all reasonable likeli-
hood), the average for the 3878 companies should certainly lie
between

£[4-7435±3(0-122)].

The true average was found by actual calculation to be £4-779,
well within the above limits, although the original items varied from
nil to £103, being grouped according to the nature of the security
— Government, Railways, JMines, etc., etc., and the averages and
S.D.'s on successive pages differed materially. This aggregation,
Bowley remarks, is very similar to that found in wages in different
occupations and localities, and in man\^ other practical examples.

The value of the second experiment due to Dr. Bowley lies in the
suggestion that similar means can be ajDplied with good results to
the investigation of many social phenomena.

If out of a large group a comparatively small sample of statistics
is collected in the purely random manner already described, we are
able by such means to estimate what is the average, and even to
obtain limits between Avhich the average A\ill almost certainlj- lie,
in the large group based upon values found for the average and
S.D. in the small sample.

Example (7). — With the collaboration of Mr. Burnett-Hurst and
a number of other workers, Dr. Bowley conducted an inquiry into
the conditions of working-class households in four representative
towns — ^Northampton, Warrington, Stanley, and Reading — the
results of which are published by Messrs. Bell and Sons under the
title of Livelihood and Poverty. They are similar in character to
those obtained by Rowntree in his study of conditions in York,
but what is peculiar to Rowley's inquiry is that only a sample,
about 1 in 20, of the working-class houses in each town was
examined, and the conditions in the towns as a whole were deduced
from these samples.

We are not concerned here with the actual facts disclosed by the
investigation, striking as they are, but with the explanation of the
sampling method adopted, and as to that it may be remarked that
the foundation on which it rests is precise^ the same as that which
underlay the example of the 999 black and white sheep. The
main point to notice here again is that Bowley was careful to select
his samples in unbiassed fashion as follows : ' For each towni a list
of all houses . . . was obtained, and without reference to anything

174 STATISTICS

except the accidental order (alphabetical by streets or otherwise)
in the list, one entry in twenty was ticked. The buildings so
marked, other than shops, institutions, factories, etc., formed the
sample.' It will be evident that tliis method of choice is not quite
on the same level of randomness as that followed, for example, in
drawing cards from a w^ell-shuffled pack, each card to be replaced
and the pack reshuffled before the next is drawn ; but, for that
very reason, the results of the experiment are all the more Hkely
to be well within the limits of error provided by the formulae of
the ideal case. The deliberate selection of every twentieth house
in each street is likely, that is to say, to give a more representative
picture of the tovm. as a whole than would be obtained by selecting
the same number of houses in a purely random fashion which might
by chance give too much emphasis to some street or district.

A practical test of the goodness of the sample was possible by
comparing the results in a few instances with information availa])le
from other sources. In order to make the method of working
quite clear, let the guiding principle first be recalled : —

' If , in a random sample of n items, the proportion of successes
is p, then the proportion of successes in the universe from which the
sample is selected will not be hkely to fall outside the limits

p±3(0-6745)V(i3?/w),
and, if that universe contains altogether N items, the number of
successes will not be likely to fall outside the limits

Ni)±3(0-6745)NVMw)-'

In Reading the total number of all inhabited housea in the
borough was 18,000 at the time of the inquiry, i.e. N^ 18,000.
The total number of houses visited was 840, i.e. ?i=840. If we
call a house assessed at £8 or less a ' success,' the number of such
houses found in the sample was 206.^

Thus 2^=206/840, g= 634/840,

and the number of houses rented at £8 or less in the whole borough
should be

N^ with a p.e. = 0-6745N\/ (^0-/71)
i.e. 4414+180.

The actual number of houses so rented was known from other sources
to be 4380, well within the limits forecasted.

The value used for p in the above is that given by the sample,
but when we know the actual number of successes in the universe

rURTHER APPLICATIONS OF SAMPLING FORMULA 175

as a whole, as in this case we do, we might use the true vahie of
p, i.e. the value for the universe in place of that for the sample.
The argument might also be put in another way without affecting
the principle employed, thus : —

The number of houses rented at £8 or less in the whole borough
was 4380.

But the proportion of houses sampled in the whole borough was
840/18000, i.e. 1/21-43.

Hence the number of houses at the above rental to be expected
in the sample^ 4380/2 1-43=204.

The number actually found in the sample was 206, with a probable

=0-0745 V(840 X r%\% X \W%^)
= 8, approximately.

Again, the number of persons engaged in a certain occupation at
Reading was known to be 761 in the borough as a whole. Hence
the number of persons so engaged to be expected in the sample
was 761/21-43, i.e. 35.

The number actually found in the sample was 29 with a probable

®^'°^ = 0-6745 V(w;>^)

=0-6745 V(840 X tUU X kiUl)
=4, approximately.

Further examples of the method are here given, in each of which
the total number of events is small so that the number in each
sample is also small, and since, as we have seen, the accuracy or
precision of the proportion of successes discovered in any sample
varies directly as the square root of the number of events the sample
contains, the results cannot be expected to be so good when this
number is small.

Example (8). — 514 candidates sat a certain examination paper ;
their marks ranged from 3 to 64. The candidates were numbere<-l
consecutively from 1 to 514, and a random sample of 90 (17i per
cent.) was selected from among them by writing down the 90
numbers formed by the digits in the seventh decimal place, taken
in groups of three, in the logs of the numbers 10104, 10204,
10304, . . . , as given in Chambers's Tables, neglecting all numbers
greater than 514 and calling such numbers as 005, 037, etc. — 5,
37, etc. In this way each of the numbers between 1 and 514 stood
an equal chance of inclusion.

176

STATISTICS

The distribution of candidates in the sample is compared with
that for all 514 together in the following table : —

Percentage of All

Percentage of Candidates in

No. of l\Iarks Obtained.

Candidates who obtained

Sample who obtained

these Marks.

these Marks.

p.e.

Less than 15

8

8±l-9

15 but less than 25

19

17±2-6

25 „ „ 30

16

18±2-7

30 „ „ 35

18

13±2-4

35 „ „ 40

15

17±2-6

40 „ „ 50

19

18±2-7

50 and over.

7

10±21

The reader might verify the p.e.'s given in the last column :
e.g. proportion in the sample obtaining less than 15 marks=7/90 ;
therefore ^^=7/90, g= 83/90.

Hence the S.D. for this group

-V[7(1-9V)]
-:2-54,

and the S.D. for the percentage

= -V(rX2-54-2-8.
Thus the p.e. for the j)ercentage

= ^a— 1-9, approximately.

Exami)le (9) deals in a similar way with the data concerning
infectious diseases in 241 towns in England and Wales previously
recorded on p. 62.

A sample of 60 towns, i.e. about 25 per cent., was chosen in a
random fashion as in the last example, and the sample distribution
is compared below with that of the 241 towns as a whole.

The verification of the probable errors in this and the next case
is left to the reader.

Case Rate per 1000
of the Population.

Actual No. of
Towns so rated.

No. as suggested by
the Sample.

1 and under 5
5 „ 9
9 „ 13
13 and over.

85
86
42

28

p.e.
92 ±10
96±10

28± 7
24± 6

FURTHER APPLICATIONS OF SAMPLING FORMULA 177

Example (10) is concerned with the annual output j)er head in
142 different types of employment as given in 1907 by the Census
of Production [data from Sixteenth Abstract of Labour Statistics of
the United Kingdom, Cd. 7131]. The distribution suggested by a
random sample of 50 different occupations is compared with that of
the complete list of 142 occupations.

No. of Occupations

No. in Complete

Actual No.

Output i)er liead.

in Sample with

List as deduced

found in

this Output.

from Sample.

Complete List.

p.e.

Under £60

4

ll±3-6

12

£60 and under £80

16

45 ±6-2

42

£80 „ £100

6

17±4-3

25

£100 „ £120

10

28±5-3

20

£120 ., £190

8

23±4-9

27

£190 and over

6

17±4-3

16

The S.D. in each of the last three examples has been calculated
by using the value for p given by the sample, which is the value
one must fall back upon in practice when the true p for the whole
distribution is unknown. In any case where ^^e are able to test
our sample by comparison with the whole distribution, however,
it is possible to use the true value of p, e.g. in Example (10)
output £100-120, 2)=-20/142 as opposed to 10/50.

M

CHAPTER XV

CURVE FITTDSTG PEARSON" S GENERALIZED

PROBABILITY CURVE

It may be recalled that in the introductory chapter an outline A\as
given of the manner in which the theory of Statistics might be
conceived to develop. It was shown how the desire for simplifica-
tion and the need for compression leads to the division of a large
mass of figures dealing with any given matter into groups ; indeed,
it may well be that the statistics have been so arranged at the
source in the act of collecting : e.g. we may have to deal with
so many males of height 54 in. and less than 55 in., so many of
height 55 in. and less than 56 in., so many of height 56 in. and less
than 57 in., and so on. Here corresponding to each given height,
which we maj^ label x, or each range of height, such as x^ to Xg,
we have a certain frequency' of males of that height or range,
which frequency we maj^ label y, and hence a frequency table can
be formed showing the variation of y with x. Further we have
seen how such pairs of corresponding values of x and y can be
plotted so as to picture the complete observed frequency' distribution
to the eye.

Now the representation thus made, though helpful up to a point,
is not entirely satisfactory. Whether we simplj'^ join up successive
points (.r, y), or set up rectangles of varpng height y on bases
spanning the successive ranges of x, or erect ordinates (y's) at the
mid-points of these bases, joining the summits in the manner
previously described, the connection so established betAveen each
observation and the next is too superficial, depending merelj^ on
the fact of casual neighbourship, and may sometimes give a false
impression of frequence' and changes in frequency in the population
of which the observations are but a sample. And this is neces-
sarily so if we confine ourselves strictly to the data observed.

One difificulty which has to be faced is that only "svithin certain
broad limits can we trust our observations to give us information
which is trul}^ representative of the population in which we are

178

CURVE FITTING 179

interested. We seldom if ever deal with the wliole poiiulation :
in fact it maj' be so large that it is impracticable even to reckon it ;
instead we make a random or unbiassed selection of a smaller but
adequate number of individuals belonging to the population, and
classify them according to the size or nature of the character which
concerns us. But, granted that our sample is adequate in size
and unbiassed, the numbers obtained in the different groups of the
frequency distribution will still be subject to the errors of random
samj)ling, and it is only after these errors have been calculated that
we can lay down the probable limits within which our sample may
be regarded as really representative of the population as a whole.

Another difficulty arises owing to the fact that our observations
in general do not cover the whole field of values of the variables x
and ?/ ; we may quite Ukely M^ant to know the percentage frequency,
y, of individuals with a character (height or whatever it may be) x
which does not chance to be any one of the x's observed, if the
observations are only recorded according to discrete (separately
distinct, like 5 ft., 6 ft., 7 ft.) values of x ; on the other hand, if
the observations have been classed in groups, the frequency in
which we are interested may refer to an x which does not coincide
with the centre of an}- group or which is even outside the range
altogether. We have therefore further to inquire whether such
information can be deduced in any waj^ from the statistics collected.

Now it so happens that both these difficulties disappear if we
can only attain the ideal already outlined in discussing graphs,
and find a suitable curve to ' fit ' the statistics observed. 8uch a
curve would not necessarily pass through all or any of the points
[x, y) representing the observations, for these, as we have remarked,
are subject to errors of random sampling and the observed frequency
y of any x may be greater or less than the corresponding y in the
population at large to which the curve is presumed to approximate.
The curve in short must remove the roughnesses which are in-
separable from ordinary observation. Moreover, given any x, not
merely one of the x'a observed, it must be possible to read off from
it the corresponding y, the frequency appropriate to that x.

It is not always accurate enough for our purpose to draw a curve
by 6ye> passing as evenlj^ as possible through the middle of the
points observed in the manner conceived in an earlier chapter. It
is necessary in some way to find an algebraical formula, possibly
even a trigonometrical, exponential, or more complex expression,
which will give the y corresponding to any x desired. This formula
or equation must depend upon the statistics collected : i.e. the

180 STATISTICS

constants involved in it must be directly and fairly easily computed
from the y'a observed, and the results of all the observations should
enter into the equations wliich determine the constants in order to
make use of the full information at our disposal. In addition, the
method of determining the equation and its constants should be as
general as possible, so relieving us of the trouble of discovering a
new method owing to the failure of the original one at nearly every
trial. Finally, the equation should not be so intricate as to make
the labour of calculating y for any given x too heavy to be attempted
with the ordinary equipment at the statistician's disposal. Once
such an equation is found it is a fairly straightforward proceeding
to trace the curve for which it stands, and it will remain after^\'ards
to test the goodness of fit in some more refined way than by seeing
how closely it passes through the observed points by eye.

When we come to review the shapes of the frequency polygons
or histograms most commonly met, we find that the majority

of them start from low fre-
quency, rise to a maximum
as X. the character observed,
increases, then fall again to-
M^ards zero very likely at a

Fm. (27). . ,

different rate. In fact the
statistics suggest a shape something like that shown in fig. (27)
for the corresponding frequency curve, though we cannot be sure
that it would coincide -with the axis at either extremity. [Cases
do occur where the curve has two or even more humps (maxima),
but we purposely restrict ourselves to the simpler and more frequent
tj^pe described.]

Now the simplest shape to deal with from the algebraical point
of view would certainlj^ be symmetrical in character, corresponding
to statistics ^^'hich rise and fall at the same rate, though this would
not necessarily be the most common shape among the records of
actual life. In order to simplify our i^roblem, therefore, we might
start by making up for ourselves an ideally simple set of statistics
which are jierfectly symmetrical, and see whether we can discover
a process for fitting a curve in a case of that kind. If this prove
successful it might be possible afterwards to adapt the same process
to an unsymmetrical or ' skew ' set of statistics made up in a similar
way. Then finally we should inquire whether actual observations
conform to any of the types of curve discovered, and, if so, how
they can be fitted together.

Now in manufacturing our statistics we must keep before us the

CURVE FITTING 181

object at which we are aiming. Given the statistics, what we
want is a formula, algebraical or of some other kind, to fit them.
This raises the possibility of choosing the statistics themselves in
some algebraical form, and such a form is at hand in the binomial
expansion, which is, in fact, one of the first examples of a general
symmetrical expression one meets. Thus

(a+6)2=a2+2a6+62
(a+6)3=a3+3a26+3a62+63
(a+6)^=a'^+4a36+6a2624-4a63+64
(a+6)5=a5+5a^6+10a362^10a263+5ai^+65

(a+6)«=a«+wa«-i6+^^:^^'-^V-262+ .

1-2

n{n- 1 )^26n-2^ ^„6^-i+ 6".
1-2

Clearly all these expressions become perfectly symmetrical if we
put a=b, for they read the same whether we run from left to right
or from right to left.

We have already seen what an important part the binomial
expansion plays in the early stages of the theory of probability :
e.g. (I+I)^". when expanded, tells us at once the proportion of times
on the average we may expect 10 heads, 9 heads and 1 tail, 8 heads
and 2 tails, and so on, when we toss an evenly-balanced coin ten
times in succession ; or again, if p is the probability that a certain
event will happen, and q the probability that it will fail to happen
at one trial, then the probabilities that it will happen p times,
(^—1) times, {p—2) times, ... in w trials are given by the succes-
sive terms in the expansion of ip-\-qY- However, we make no
assumption for the moment as to the values of a and 6, except
that in the symmetrical case with which we begin they are equal,
and we have as the successive terms of (a+^)" '■ —
n{n—l]

a'\ na'

1-2

Let us suppose that our observed statistics take the above form
so that these terms may be plotted as a succession of ordinates,
i/v y-i' 2/3» • • • > Vn+v associated with abscissae, x^, .Tg, x^, . . . , x^+i,
at equal distances apart measured, say, by c ; for convenience we
may place the origin as in fig. (28), so that

Xi=c, X2=2c, x^=Sc, . . . , a;„^j=(?i-j- i)c,

182 STATISTICS

and we can then form a frequency polygon, where

■-re, ijf-

n{n~\){n-2)

1-2-3 . . . (r-1)

(n— r+2)

are typical values of a pair of the variables x and y, each such
pair defining a vertex of the polygon.

Now in this case, since the statistics have been artificially built
up by ourselves and are not in reality a random selection, they are

Fio. (28).

not subject to errors of sampling and the fitting curve should,
therefore, pass through the summits of all the t/'s, or, perhaps
better, touch each of the lines joining adjacent summits. The
curve only differs from the neighbouring outline of the polygon in
that the latter is discontinuous, it alters its direction relative to the
axis of X by jerks at equal intervals c measured along OX, whereas
the former must rise gradually and continuously and then fall in
the same way. This is one sense in which we mean that the fitting
curve removes the roughness of the observation statistics — it gets
rid of jerks besides filling gaps in the observations.

It will be clear that as n increases and c diminishes (and this is
what we aim at in collecting statistics, though it has not been assumed
in what immediately follows) the discontinuity in the polygon
becomes less and less pronounced and the outline of the figure
approximates more and more closely to the
curve. Moreover this approximation gains in
intensity if we make the slope of the curve at
each appropriate i^oint the same as the slope
obtained by joining up the summits of adjacent
ordinates of the polygon.

(yr^.-yr)

Now the expression

{yr+X-VrMo

CURVE FITTING

183

is the measure of the gradient from the rth ordinate to the (r+l)th,
and

yr+x-yr_0''

c
c

= 2/r

n{n—l)

{n—r-\-l) n{n—l)

w(n-

1-2

■1)

(w— r+2)

1-2
-2r+l

(r-1)

1-2

r+1

(n-r-f2)"|

re

If this be also taken as the gradient of the tangent to the curve at
the point midway bet\\'een (x^, 2/r) ^^^ i^r+v Vr+iit calling this point
{x, y) we have, since, in the notation of the differential calculus.
dy

dx

is the measure of the gradient of the curve at this point,

dy Vr+i-yr

dx

=yr

n—2r-\-l

re

And

.T=i(.r,+z,+,)=K^c+(r+l)c]=---(2r+l)

y=l(yr+y,+i)=^^ ^- ^^

{n-r+2)

1-2

Hence
n-

yr—

Thus

2?-+l 2ry

. (r-1)

(n+2)-{2r+l)

r+1

^1^(^ + 1)-
2r

2^

re

?i+2— —
(w+l)c\ c

re n-\- 1

c?a: (n-l-l)c\ ej

But if M'e had started with any other two adjacent ordinates
instead of y^ and y^^-^^ we should have been led to exactly the same
relation connecting tlie corresponding x and y of the required
curve, for r, which serves to particularize the ordinates, does not
appear in the relation at all — their individuality has been eliminated.
The above equation may thus, if we please, be taken as holding
good for, and therefore defining, all pomts {x, y) of the fitting curve :
it is, in short, the differential equation of that curve.

The equation may be slightly simplified by transferring the

origin to the point (w+2)- . 0 , evidently the point 0' in fig. (28)

184 STATISTICS

corresponding to the maximum ordinate of the polygon or curve.
Algebraically, this merely means that for x we must WTiLe

[(n-\-2)c~]
x-\-- ~ in the equation, which then becomes
2 J 1

dy 2y f 2x\ Axy

dx (w+l)c\ c/ (w+l)c2

We may pass to the equation proper of the curve by integration.
Thus, separating the variables,

lxdx=0.

J y (n+1)

2x^
Therefore, log t/+-^-— +A=-0,

{n-\-l)c^

where A is a constant.
Hence y=y^e'-''''"<''+'^\

where ?/q is a new constant.
This may be WTitten

y_y^e--/^-, . . . (1)

where a^={n-\-l)c^j4:, and it is called the 2)robability curve or normal
curve of error*

Let us now see whether the ^jrocedure so far followed is applicable
in the case of an unsymmetrical or skew distribution of statistics.
With this object we will suppose the frequencies of observations in
successive groups to be represented by the corresponding terms in
the expansion

and as before we can form a frequency polygon by joining the
summits of the ordinates

n(n—l) „ „

yi=p''> yt=nv''-% 2/3=^-^~i^ '"-r. • • • . yn+i='t,

[* Karl Pearson's method of getting the normal curve equation has been
adopted as tlie basis of the above discussion, in preference to that usually
followed, which develops the curve also from the binomial expression but some-
what on tlie lines of Laplace and INjisson. They showed that the sum of all the
terms lying within a range t on either side of the maximum term in the expan-
sion of (p + q)" is approximately

1

^/2w■<7

■ [+t
J-t

where (r= s,l{n2)q), whence the equation of tlie curve is derived. (See Historical
Note at the end of Chapter xviii. )]

CURVE FITTING

185

erected on the axis of x at distances from the origin given by

the figure being very similar to that in the symmetrical case.

The gradient of the fitting curve where it touches the join of
(^r> Vr) to (^,+1, Vr^y) is given by

dx c

and we must try and express the right-hand side as before in
terms of {x, y), the co-ordinates of the mid-point of the line joining

(^r. y,) to (XV+I, yr+x).

We have

dy_\
dx c

'n{n-\). . . in-r+l)^n-Y_ ^(^-1) • • • (^-^+2) n_,.+Y.

1-2

1-2 .. . (r-1)

■1

^n_Y-i n{n-l)

(w— r+2)

1-2

(r-1)

n—r-\-l

q~p

Also

2x=x^.-\-Xj.^i=rc-\-{r-\- l)c=^{2r-\-l)c
?i('/i— 1) . . . (/I— r-f2)

2y=yr+yr+i = -^r^^^ , — 7^ — • p

1-2

Thus

dy_2y/n—r-{-l
dx c\ r

q-p

)/(

(r-1)
w— r+1

n-Y-i

7i—r-\-l ,
q^p

r

g-^p

=^[{n+l)q-r{p+q)]![{n+l)q+r(p^q)]
c

2y,

^[2{7ii-l)qc-ip-{-q)i2x-c)]![^in-i-l)qc+{p-q){2x-c)].

This, being true for all such pairs of values of x and y, is now in a
form independent of any particidar point on the curve we seek ;
in other M'ords. it may be taken as the differential equation of the
curve, and it is evidently of the type

dy ia—x)
dx {p-\-yx)

where a, /3, y involve only p. q, n, etc., the constants of the distri-
bution we set out to fit.

-' y ■'yx-\-h

186 STATISTICS

The equation is simplified if we transfer the origin to the point
(a, 0), when it becomes

dy^ yx _^
• dx yx-\-h
where 8=j3+ya.

To integrate, separate the variables as before :

{dy
y

Therefore,. log y+lf?Z^±^~^dr=0

yJ y.r-j-S

log 2/+^-- log (ya-+S)+A=0,

y y

where A is a constant,

or j/=Be-^^^(y.T+8)%

where B is a constant.
It may be written

y=y„e-''^(l+^V' . . . (2)

where A;=l/y, a=8/y, and y^ is a new constant.

This, then, may prove a suitable type of curve to fit a set of
statistics forming a skew frequency distribution, but the question
now arises whether equations (1) and (2) are the most general
types possible. Clearly (1) is only a particular case of (2) obtained
by making 2^=9, and, this being so, (2) may itself be a particular
case of some still more general type.

Light may be thrown on this if we consider the geometrical
bearing of the differential equation obtained in the last case :

dy yia—x)

dx j8+y.r

(3)

The presence of y and {a—x) in the numerator of the right-hand

dy
side of (3) shows that — vanishes when y=0 and when .r=-a, i.e. the
dx

curve touches the axis of .r ^^hf ro the two meet and there is a

maximum point on the curve at .r=^a. (Since a is the particular

value of the organ or character x for A\hich the frequency is a

maximum, a is of course the mode.) Now these two characteristics

are the very ones to which we a\ ished to give symbolical expression

since they serve to describe in broad outline what was agreed to

CURVE FITTING 187

be the trend of the majority of frequency distributions — the rise
from zero to a maximum, at first gradually, then faster, and, after
passing through the maximum, the fall to zero again, generally at
a different rate.

As to the denominator of equation (3), the corresponding equation
for type (1 ), before the origin was changed, was similar to equation (3),
except that it contained no x term in the denominator, and that is
readily understood when we note that y is a multiple of (p—q)
and thus vanishes when p—q. Now, if from (3) we get a less
general type of curve by dropping the x term in the denominator,
we may perhaps get a more general type by adding an x^ term, and
even an x^ term, an x* term, and so on. In fact there seems no
reason why the denominator should not be any function of x, say
f{x), which, however, we shall suppose for simplicity capable of
expansion in a Maclaurin's series of ascending poAvers of x which
converges quickly.

We are led to propose, therefore, as more general than C3), the
differential equation

dy_ yjx+b) ^ ^
dx px'-\-qx^r

We stop at X' in the denominator because it has been found, if we
may anticipate results to save needless labour, that beyond this
point the heaviness of the calculation involved and the decreasing
accuracy of the higher moments that have to be introduced out-
weigh any other advantage gained. The curve or set of curves
resulting from the integration of equation (4) is knoAATi as Karl
Pearson's Generalized Probability Curve, and their author has
stated that, while it comprises the two other types as special cases,
it practically covers all homogeneous statistics he has had to deal
with.

Just as the differential equations in the first two cases considered
were related respectively to the s3aiimetrical and the skew binomial
expansions, so is equation (4) related to the hypergeometrical
expansion

the successive terms of which express the probability that r black
balls, (r— 1) black balls and 1 white ball, (r— 2) black balls and
2 white balls, . . ., r white balls, will be drawn from a bag contain-
ing pn black balls and qn white ones, where {p-\-q)=^\, when r balls
are drawn in all, each being replaced before the next is drawn.

188

STATISTICS

If the terms of this expansion are represented by ordinates of
which the summits determine a polygon as in the binomial eases,
the corresponding expression for the gradient of the curve at any
point is given by an equation of type (4). We need not go over
the detailed proof of this statement since it follows precisely the
same lines as in the previous cases.

The method of integration of the equation

dy_ y{x-\-b)
dx px--\-qx-\-r

depends upon the nature of the roots of the quadratic in the
denominator which may be Avritten

px'-\-qx-\-r=p

\4p'^ pi _

=P

[(-si-

4r2 q^ / 52
q- 4:pr ^ ipr

=P

[(-4)"-

4r'

'

where K^q-j4pr, and it is evident that the quadratic splits up into
real factors if /c(/c— 1) is positive. This is the case when k has any

negative value, or when it is positive
and greater than 1, the truth of which
may be seen more efifectively if the
curve

?/=«(«— 1),

Y

II

K-

/ K-*r(>\)

o\

/K=1 k

K=

0^

^

Fia. (29).

IS

a parabola symmetrical about the line
K=\, be drawn, fig (29), by plotting
y against k.
Further, the product of the roots of the quadratic

px--^qxA^r=^^

r_4/-- cf' _4/--

p q- 4pr q-

so that the roots when real will be of the same sign if k is positive
and of opposite signs if k is negative. The boundary lines

/c=0 and « = !

thus divide the whole field into three parts, as shown in fig. (30), in
one of which the roots are real and of opposite sign, in the next

CURVE FITTING

189

the roots are imaginary, and in the third the roots are real and of
the same sign. At the boundaries we get particular cases as
follows : —

K=0 : this requires q=0, since /c=q-l42)r, which makes the
roots of the quadratic equal but of opposite sign, unless ^=0 also,
and in that case both roots are
infinite ;

« = 1 : tlie roots are real and equal
and of the same sign ;

« = oc : this requires p =0 or r =0 ;
in the former case one root of the
quadratic is infinite, and in the
latter one root is zero.

Y

<*-,

=n

2

/

'k> \

-

13

/

^ \

c

.tl

v

h

=.A

II

^

>^

v^

o "^

OD

■~

c

e eo

\ ""^

^

/

II

to g

\a:

/

Z c

0=

/

0

o

/

IT

K

Fio. (30).

Thus, returning to the differential
equation, the curves which result
from the integration

fch/ f {x-[-b)dx

J y J px--\-qx-\-r

are of different types according to the value of /c, which is therefore
called the criterion.

Type I. — K—^'. Roots of px--\-qx-{-r—^ real and of ojrposite sign.
In this case we may write

and so get

]}X~^qx^r^^p{x-\-a'){x~^')
{X'\-b)dx

; y .' via ■

^0,

p{a' -^x){^' — x)
or, transferring the origin to the point ( — 6, 0), the mode, we have

(dy f xdx

J~y Jp{a-b-]'X){fi'-\-b-x)

'dy . I xdx

=0.

or

J y J i){a

where
Therefore,

lo

+x){^~x)
dx . 1

^0,

1 ; a ax I I

pJ a+x a-\-B pJ I

^ dx

a=a

1

pJ a+x a+j8 ' pJ ^—x a+/3

where A is a constant.

A=0,

190 STATISTICS

Thus log y=^-^—^a log (a+a-)+iS log (^_;r)]+log B,

where B is a constant,

whence y=B(a+.r)p(«+«(^— a;)''("+^)

i.e. yM'+a) {'-^) ■ • ■ f'^'

where i/ = l/p(a+^) and y^ is a new constant.

This is a skew curve of limited range, bounded by the lines .r =— a
and a;=+jS, with the mode at the origin.

Type II. — «:=0. 9=0, but not p=0. Roots of 2JX^-{-qx-]-r^0
equal and of opposite sign.

This curve is just a particular case of type I., Avhich reduces to

/ x^ \ ^"^

y=yo 1---2 . • • • (6)

symmetrical about the axis of y (because for any value of y there
are two values of x, equal and of opposite sign) and of limited
range bounded by a: =— a and x = -\-a, with the mode at the origin.

Type III. — K=oc.* p=0, but not r = 0. One root of2)x'^+qx-\-r=0
infinite.

This is the skew binomial case over again. It may be also de-
duced from type I. by making one root, say jS', tend to infinity.
The curve then takes the form

because j8=^' + 6, so that ^ tends to infinity with /3'. Hence

l\x-

a/ x^co

'+A

where A=— ^/x.

Thus y=y (l+^y^e"", ... (7)

\ a '

a skew curve limited in one direction by the line x=—a, with the
mode at the origin.

[* Although theoretically this type corresponds to an infinite value for k, in
practice it will as a rule give a reasonable fit provided k is numerically greater
than 4. (See W. P. Elderton's Frequency Curves and Corrdation, p. 50)].

CURVE FITTING 191

Ty2>e IV. — k -]-''"' and < 1 . Boots of px^ 4- g-a; -f r = 0 imaginary .
Put «(«— 1) = — A", and the differential equation then leads to
rdy r {x-\-b)dx

J u

Transfer the origin to the point ( — — , 0

X'{-b—-' ]dx
2p/

log ,==.4+^1 log (.=+4'!iV(^ -/-,)/, tan- ^
2p \ q- / \-p 2p'/2rA 2rA

where A is a constant.

X2\-"> -,-tan

Therefore, y=yj If „ e * ... (8

a

, 2rA 1 If, q

where a= — , m —— — , v =— — o— -^

q 2p ap\ 2p

and ?/o is a constant.

This is a skew curve of unlimited range in both directions. The

position of the mode is found by putting — =0 in (8) after differ-

dx

entiation, or, what comes to the same thing, is seen by direct refer-
ence to the differential equation itself. Thus the distance of the
mode from the origin

Type V. — K=\. Roots of px^-\-qx-\-r=0 real and equal.
The equation to integrate becomes

{dy f {x-'rb)dx

lU

' '^<

192 STATISTICS

Transfer the origin to the point I — — , 0 ), and this becomes

y J ^j,r-

log t/= A+- log x-^.ib- ,

p p \ 2pjx

where A is a constant.

Therefore, y--^tj^x^''Pe ^'^ ^^'^^

y--yoX-'e-v/^, . . , (9)

where s — — !/}), y= ( 6 — — ), and ?/(, is a constant.

Here x cannot become negative, so that the curve is skew and
limited in one direction. The distance of the mode from the origin

Type VI. — K-{-'" and >1. Roots of ■px-'^qx-^r=0 real and of the
same sign.

Equation becomes

fdy I {X'^b)dx

y J p{x-^a){x+^)

logy =

b-a 1 (6-iS) 1

jpi^—a) .T+a l'>{a-~fi) x-\-fi_

dx

=A+— L_[(6-a) log (.t■-^a)- (6-/3) log (.r+^)].
p(^-a)

where A is a constant ;

or, transferring the origin to (— jS, 0),

log 2/=A+~^^^[log j.r-(^-a)p-log x^^

y=yo(x-ar-x-^', • • . (10)

where a=:^-a, qz^{b—a)lp{^—a), qi = {b-^)ip{^—a), and y^ is a
constant.

This is a skcAv curve bounded by x—a in one direction. The
distance of the mode from the origin^ — (6— j8)=agj/(g'i— ^a)-

CURVE FITTING 193

Type VII. — K=0, q=0,p=0. Roots of the quadratic px--\-qx+r=0
both infinite.

This is the symmetrical binomial case over again and the integra-
tion reduces to

fdy_
■' y

J^-'d.,

J r

or, transferring the origir

I to (-

b, 0),

1 y

-\>

logy--

=A+ ^x\
2r '

where A is

a constant.

Therefore

y-

=yoe

. (11)

where y^ is a constant and ct-= — r.

This curve, the normal curve of error, is symmetrical about the
axis of y, where mean and mode coincide, and it is of unlimited
range on either side of it.

K

CHAPTER XVI

CURVE FITTING (continued) — THE METHOD OF MOMENTS
FOR CONNECTING CURVE AND STATISTICS

We have now completed the first stage of the discussion upon which
we embarked : we have found by the apphcation of general prin-
ciples various types of curve, represented by different equations,
which are said to fit more or less satisfactorily a considerable number
at all events of frequency distributions composed of homogeneous
material.

Our next task is to pass from the general to the particular, to
see how to set up a connection between an actually observed fre-
quency distribution and the appropriate theoretical curve. This
again seems to break up into two parts — (1) to find a way of deciding
which type of curve to adopt in a particular case ; (2) to determine
the constants of the curve in terms of the observed statistics ; but
since the criterion, k, which distinguishes one type of curve from
another is itself a function of the constants of the curve before
integration, it follows that the solution of the first part is incidental
to that of the second.

The general method proposed for determination of the constants
of the curve in terms of the observed statistics is the now well-known
method of moments due to Karl Pearson, whereby the area and
moments of the fitting curve are equated to the area and moments,
calculated from the statistics, of the observation curve.

If a frequency table be drawn up (see Table (40)) showing the
number / of observations corresponding to the deviation x of each
value, or group mid- value, X of the character observed from some
fixed value, the expression

^'Ji^^-Ji+ ■ ■ ■ -h^rfr+ ■ • •
is called the first moment of the distribution with reference to the
fixed value, which may be termed the origin. Similarly,

^Vl + ^"2/2+ • • • +-^'-r/r+ • • •

is called the second moment, Ux^, the third moment, Ux*f, the

194

CURVE FITTING

195

fourth moment, and so on. The following notation will be found
convenient for working purposes :—

vi ^ Z"/ ' '' ' N 2"/ ' * * *

Undashed letters are reserved for use when the distribution is re-
ferred to its mean as origin, in other words when the deviations of
the X's are measured from the mean X.

Table (40).

Deviation. Frequency.

First
Moment.

Second
Moment.

Third
Moment.

Fourth
Moment.

X,
Xj

A

'fr

^-rfr

^2/2

^\fr

Totals .

N

N'l

N'2

N'3

N'4

Now each N in the frequency table is the sum of a number of
discrete quantities which only tend to form a continuous series as
the class intervals are made very small and the number of observa-
tions is made very large. The corresponding frequency polygon
or histogram, if we drew it, would at the same time tend to become
a continuous curve, the observation curve. If that limiting stage
were attainable, if we could actually get an infinitely large sample
of observations in which the character observed changed by infinitesi-
mal amounts, we could then replace the isolated /'s of observation
by the corresponding ?/"s, the ordinates of this observation curve,
and to get the moments we could write instead of the discrete
sums

Ef, Sxf, Ex'^J, . . .,

the continuous integral expressions

jy'dx, jxy'dx, jx~y'dx, . . .,

taking in the whole sweep of the curve by integrating throughout

196 STATISTICS

the range of deviation x. We should then have, if areas and
moments are equated according to Pearson's method,

jydx=jy'dx, jxydx=jxy'dx, jx^ydx=jx-y'dx, . . .,jx"ydx=jx^y'dx,

where y is the ordinate of the fitting curve corresponding to the
ordinate y' of the observation curve.

In practice, however, it is impossible to go to this limit : we
cannot deal with an infinitely large sample, so we take as large a
sample as is convenient, calculate the rough moments, N, N'^, N'2 . . •,
and find approximately what corrections or adjustments are neces-
sary to obtain the moments of the observation curve, a procedure
which is really equivalent to the determination of the area of a
curve when only a number of isolated points thereon are kno^sn.

For the full analytical justification of the method of moments
the reader is referred to Professor Pearson's original paper. On
the Systematic Fitting of Curves to Observations and Measurements
[Biometrika, vol. i., pp. 265 et seq. ; also vol. ii., pp. 1-23], where
it is shown that ' with due precautions as to quadrature, it
gives, when one can make a comparison, sensibly as good results
as the method of least squares.' The latter, which is the traditional
way of approaching all such problems, is shown to be impracticable
in a large number of cases, either because the resulting equations
cannot be solved, or, when they are capable of solution, because
the labour involved would be colossal.

Let us consider next how to deduce the area and moments of the
observation curve from the statistics, in other words how to get

jy'dx, jxy'dx, jx^dx, . . .,

the integrals being taken throughout the range of the curve, when
we know the frequencies corresjaonding to only a certain number
of values or elementary ranges of the deviation x.

Now the character observed may be capable of the deviations
actually recorded and of no values in between, e.g. measuring
deviations from ' no rooms ' as origin, we might have /^ one-roomed
tenements, /g two-roomed tenements, /g three-roomed tenements, but
there could be no such thing as a two-and-a-half or a three-and-a-
quarter-roomed tenement ; on the other hand, any recorded devia-
tion, Xf, may be only the mid-value (used as a convenient and
concise approximation) of a group of observations including all in
the continuous range from (Xj.— ^) to (Xy+^), where unit deviation
is the class interval : thus we might have /^ males deviating by
+ 6 in. from 5 ft. (comprising all the males observed between 5 ft.

CUKVE FITTING

197

5^ in. and 5 ft. 6| in.),/^ males deviating by +5 in. from 5 ft. (com-
prising all males between 5 ft. 4| in. and 5 ft. 5| in.), and so on.
These two cases must be discussed scjoarately.

(1) When the observations are centred at definite but isolated values
ofx.

The problem is to find

j.v"y'dx

(the nth moment) when we have no definite curve given but we
know the values of x and y' at a number of isolated points, say

(■^'o> y'o)^ (-^"i. y'l), (•^•2' y'2)^ • • • > {'^p> y'p)-

This is equivalent to discovering a suitable ' quadrature formula,'
i.e. a good approximation to

jzdx

<-/■

Fig. (31).

0 h \ h 2 h3 p h
{P^\]h >

Fig. (32).

in terms of knowTi points

(x-o, So)' ('^i. 2=1), (.r., ^2). • • • (''^;.> 2j,).
where we have Avritten z in place of .f"?/', and we may generally
take the ordinates to be at equal distances, A, apart. Several
such formulae have been suggested and they vary according as the
2's are situated at the ends (fig. (31)) or at the centres (fig. (32))
of the h intervals. The second type is perhaps the more useful of
the two, and we shall work out one formula in illustration of it.
Consider the first five of the given points, namely,

(.To, eo)> K. 2i)> • • • (•^4> 24)-
As a simple ' curve of closest contact ' let us find the parabola of
type

z=CQ^c^xl'h-^c.-^x-lh-^c^x'^llv'^c^x''l'h'^ . . (1)

which goes through these five points, where the c's are constants to
be determined. We may without loss of generality take the axis

198

STATISTICS

Zn Cn

24 = Co + 2Ci + 4C2+8C3+16C4.

of z to coincide with the middle one of the five ordinates, so that
the known points on the curve become

{-2h, Zq), {-h, Zi), {o, Zg), i+h, Z3), (+2;^, 24),

and on substitution in (1) we get

2^0=^0 — 2C1 + 4C2—8C3+I6C4. 2;i=Co-Ci + C2— C3 + C4.

2:3 = Co + Ci + C2 + C3 + C4.

These equations are just sufficient
uniquely to determine the c's, and
hence the paraboHc curve of closest
contact, in terms of the five given
points, but for our purpose it is not
necessary to find all the c's. Suppose
our object is to find the area of the
shaded portion of fig. (33) in terms
of the co-ordinates of the five given
points. This area

■A O +/i +2/i

• + /'/2

= / zdx

■hft

' + 1,12

~^ -h/2 ^^" "^ Ci;c/A+ c.^x-/h- + c^x^/h^ + CiXyh^)dx

+hl2
_ -hl2

CqX + Cix-I2h + C2X^I3h^ + C3xy4:h^-\-c^xy5h^
But the equations between the z's and c's at once give

22 = Co, 20 + 2:4 = 2(Co + 4C2+16C4), Zi+2;3=2(Co + C2 + C4).

Thus

Therefore

8C2+32C4 = (2o+24) — 2Z2
2C2 + 2C4 = (Zi + 23)-222

24C2 = 16(2i + 23)-(2o + 24)-3022
24C4 = (2o + 24)-4(2i + 23) + 622.

Hence, by substitution, the shaded area becomes

'•+hl2

2Zia;=^[z2+2¥sjl6(Zi + 23)— (2o + 24)-3022|

+ T9Vol(2o + 2;4) — 4(21 + 23) + 622I]

=g^^[517822-17(2o + 24) + 308(2, + 23)].

r+hi2

J-h/2

(2)

CURVE FITTING 199

these particular ordinates being appropriate when the axis of z
coincides with the z^ ordinate.
Similarly, it can be shown that

r-f-3A/2 )i

by finding the parabolic curve of closest contact through (0, z^),
{h, Zj), (2h, Z2), {Sh, 23), the axis of z coinciding now with Zq.

Now we require | zdx

(see tig. (32)), and this may be obtained by splitting up the integral
thus

+ + +...+ +

J-hl2 hhft .'bhjl •(i'-?V< ■(p-i)!'

and applying the formulce (2) and (3) to evaluate these sub-integrals.
The first and last come under head (3), while all the rest come
under (2). In fact, we fit together portions of curves of parabolic
type based on the successive groups of points

(0, 1, 2, 3), (0, 1, 2, 3, 4), (1, 2, 3, 4, 5), (2, 3, 4, 5, 6), . . .
(p-4, p-3, p-2, p-l, p), {p—S, p—2, p—l, p),

and as the points overlap, in the sense that neighbouring groups
have points in common, the curves dovetail into one another and
so provide a fairly good approximation to what we want in the way •
of integral expressions giving areas based upon the positions of
certain known points.
We have, then :—

f3h:2 Ji

zcIx=--[21zq-\-11zi + 5z,-z^]
J-h!2 24

zdx=—-[ollSz,- 17(2o-f~4) + 308(2i+23)]
i3;i/2 o / dO

i''"'\dx=^UollSz,-ll{z,+z,)-^Z()8{z,-hz,)]
hhft o760

/•9/1/2 }>

zdx=—-[o\18z,-\l{z.,+z^)^^m^^-VH)'\
.'nii 5760

j^''^''zclx=---[5ll8z,,_^- 17(2;,,_4+Zp)+308(2,,_3+Zp-i)]
.'(p-i)h 5760

■ <j)-i)h 24

200 STATISTICS

Hence, by addition,

/:

(p+i)h h

2fZx=— -[6463zo+4371zi4-6669z2+5537z3+64632„
hj2 5760

+437l2^,_i+6669v2+5537z^_3]

=;i[M220(2o+2^)+0-7588(2i+2^_i)+M578(z2+V2)
+0-9613(23+z^,_3)+ (24+25+ . . . +2^_4)].

In effect, since z—x"y', this means that to calculate the moments
from the given statistics we may work simply with the observed
ordinates or frequencies, as drawn up in Table (40), so long as we
modify the first four and the last four by multiplying them by
suitable factors. In particular, when the frequencies at the be-
ginning and end of the distribution are very small, that is to say,
when there is high contact at each end of the frequency curve,
we may dispense even with the modifjung factors also since we
may assume that before the first and after the last ordinate observed
there are others which are so small as to be negligible.

Thus, given high contact at each extremity of the observation
curve, we may write

/:

zdx=hZz,

-1x11

or, if we take the class interval as unit in measuring x so that h=l,
this gives

jyx"^dx=I!fx"^,

where the integral may now be taken as referring to the fitted
curve, since the moments of the theoretical and of the observa-
tional curves are to be equal, and the integration traverses the
extent of the curve. When, however, there is not high contact at
the extremities the same equation holds good if we multiply the
first and last of the observed /'s by 1-1220, the second and the last
but one by 0-7588, the third and last but two by 1-1578, and the
fourth and last but three by 0-9613.

In particular, when n=0, integrating throughout the curve,

jydx=i:f=l^, . . . (4)

Avhich, being interpreted, means that the area contained between
the fitting curve and the axis of x measures the total frequency of
observations, modified if necessary.

Also, when the observation moments have been adjusted, if we

CURVE FITTING

201

A;\Tite jx and fx' in place of v and v' in the notation previously pro-
posed (see Table (40)), integrating again throughout the curve,

jxydx/jydx=I!xf/'N=iJi\, . . . (o)

and the geometrical interpretation of this is that the foot of the
ordinate passing through the centre of gravity of the area between
the fitting curve and the axis registers the deviation of the mean X
from the fixed origin.

If deviations are measured from the mean of the distribution
as origin 2^(0^/) vanishes (see also Appendix, Note (5)) so that/Ai=0.

Generally, we have, ^vith the same limits of integration,

[xHjdxl\ydx =Z'a:"//N =/a' „,

and when the distribution is referred to its mean as origin the
right-hand side is written ju„.
We now pass to the second case.

(2) When the observatioyis appear in groups ranging between
definite values of x, the range of each group as a rule being the same
in extent.

Since the usual procedure here is to treat each member of a group
as though it were centred at the x at the middle of that group —
e.g. a group of school girls
each of some weight be-
tween 7 stone and 7 stone
5 lbs. would be treated as
if all its members were of
weight 7 stone 2-5 lbs. —
this case evidently reduces
to that already considered.
It is necessary, however, to
examine what correction

must be made for assum- ^

ing that all the members ^"__
of the same group have
the same x.

Consider again the expression

\x"y'dx.

The contribution to the ?ith moment coming from the z^ group of

observations (see fig. (34)) may be taken as the portion of the

h\ , / h\ ,

where

/■

Fig. (34).

above integral between limits ixQ-\-rh—-) and ( XQ-\-rh-\-

2/ V 2

202 STATISTICS

Xq is the distance of the centre of the first group from the
origin 0.

But, since all the observations in the same group are treated as
if they had the same x, by (2) this integral may be written

^[5178(.^•o+r/^)»-17}(.^o+7^/0" + (■ro+r+2^)";•

+ 30S{(.r,+~r^lhy-i-{x,-i-V+lhr}l

where /^ is the frequency of observations in the group, and this, on
expansion in powers of {xQ-\-rh) and h,

=hf,{Xo+rh) " + ^[240n{7i~imx,-hrhr-'
57 oU

+ 3n(7i— l)(w-2)(w-3)/i*(.ro+r/i)"-*+ . . .]•
When we sum for all groups, the expression

r = 0

gives evidently the 7ith moment of a set of isolated variables,
/o) /i) A' • • • fpy ^^^^ ^y Case (I) it may therefore be taken as
being practically equivalent to the required wth moment of the
observation curve, assuming that there is high contact at each end oj
the curve.

The remaining terms,

-'P Mr

mOn(n-l)h^Xo-hrh)''-^-
5760 / V 0 r ;

-^3n{n-l){n-2){n-3)h*{Xo+rh)''-^+ . . .],

may accordingly be taken as the correction required.

When ri=0, these terms vanish, so we infer, just as in Case (1),
that, when the integration is taken throughout the curve,

jydx=i:f=lSi, . . . (4) bis,

or, the area between the fitting curve and the axis of x measures
the total frequency of observations when the class interval h is
treated as the unit in measuring x.

Again, when n^=l, the corrective terms vanish, so we likewise
infer, as in Case (1), that, with the same limits of integration,

fxydxljydx=Zrf/N=fji\, . . . (5) bis,

and that /,ii=0.

When n=2, the reduction of the corrective terms gives

h-
second unadjusted moment^second adjusted momenta — ^hf,

CURVE FITTING

203

or, dividing throughout by Zhf and bearing in mind the notation
adopted with the mean as origin,

/^2 = *^2-r2' • • • (6)
when A = l as before.
When ri=3,

third unadjusted moment =third adjusted momenta - Uf,.{.rf,^rh) ;

4

but, if we refer the deviations to the mean of the distribution as

origin, 2Jf^.{xQ-\-hr) vanishes.

Therefore, 1^3=^3 » « . (7)

When 71=4,
fourth unadjusted moment

=fourth adjusted moment-} 2Jf,.{xQ^rh)--\-—Zhf.

L 80

Hence, dividing through as before by Uhf dnid taking ^ as 1,

1^4 =/i.4+ 2/^2+ 51>"

Therefore,

H-i

.-hu+±

(8)

2' 2 > 240

To sum up, the general procedure in Case (2) is to calculate
N, N\, N'2, N'3, N'4 directly from the statistics and so deduce
v' 1, v'2, v'3, v' i- Then, transferring the origin to the mean, the v's
become i/^, j/g, V3, v^ (see Appendix, Note 5), and finally the cor-
rected /i.'s are given by

/^l=0' /^2 = i'2 — i'

/^3 = I'3' ^* = '^4 — ^''2 + 240 •

These adjustments, originally due to Dr. W. F. Sheppard * [Pro-
ceedmgs of the Lond. Mathl. Socy., vol. xxix., pjj. 353 et seq.], are
applicable only when the
curve of distribution has
high contact at each ex-
tremity as very frequently
happens. To this case
w^e shall confine ourselves,
and when it does not hold
the unadjusted moments
may be used as a rough approximation failing a more refined but
also a more intricate adjustment.

The way in which the three chief kinds of average are related to

[* To obtain Sheppard's adjustments we ha^ e followed the method indicated
in Elderton's Frequency Curves and Correlation, pp. 28, 29.]

204 STATISTICS

the fitting curve is of interest and deserves recapitulation. Whether
the observations are classed as in Case (1) or as in Case (2) : —

(1) the ordinate drawn through the highest point of the curve,

since the frequency there is a maximum, fixes the modal
value of X ;

(2) the median X is determined by the ordinate bisecting the

area between the curve and axis, since there are an equal
number of observations on either side of it ; and

(3) the mean is determined by the ordinate through the centre

of gravity of the area between the curve and axis.

We have still to show how to express the constants of the fitting
curve in termsof the moments calculatedfrom the given statistics, and
it will be convenient now to make our approach from the other end.

Take the general equation of the fitting curve, express its con-
stants in terms of its moments, and substitute for the latter the
values determined from the statistics, since the basis of the fitting
is the equalization of the moments of the observational curve and
of the theoretical curve. This will enable us to determine k, the
criterion for fixing the type of curve suitable to the given distribu-
tion. When the type has been fixed it is, as a rule, not a very
difi&cult matter to express the constants of the particular type
again in terms of the observational moments.

Now the general differential equation of the fitting curve was

dy^ y{x-\-b) ^

dx px'^'{-qx-\-r
hence

j{px'-\-qx^r)dy=jy{x^b)dx,

where the integration is to traverse the complete curve.

Therefore, multiplying both sides by a;",

j{px''+^+qx''+^+rx'')dy=j{yx''+^-\-byx'')dx]

or, if we integrate the left-hand side by parts

[(^a;"+'- + gra;«+i + ra:" )y] —jy{n-\- 2^a;"+i + 7i+ Iqx" + nrx''-^)dx
=j{yx^+''^-\-byx^)dx.

But the expression in square brackets vanishes at both limits if
we suppose y to be zero at each end of the curve, so that the equa-
tion reduces to

{l+p'n^2)jyx''+^dx+{b + qu^l)jyx"dx+r)ijyx"-'^dx^0, ... (9)

CURVE FITTING 205

Now if deviations are measured from the mean of the distribution,
we have

jyxdx='Nfjii^O, jyx-dx=Nij,2, \yxhlx='i:iiJL^, etc.,

and therefore, putting n=3 in the above relation,

(l+5i))N/x,+ (6+4g)N/z3+3rN/Lt2=0 ;
put n=2, (l+42))N/x3+(6+3^)N^o=0 ;

put w=l, (1 + 3^)N/X2+^N=0 ;

put w=0, {6+(?)N=0.

Thus b = — q, and, on substitution in the other three equations, we get
5/x42>+ 3jLt3g+ 3^2r+^4 =0,

Sfx^p + r+fji,=0,

three simple linear equations to find p, q, r, the solution of which
leads to

(/ = — 6 = — /i3(/x4+ 3/A-2)/(10/x2/^4— IS/x^a— IV's).

We have thus expressed p, q, r, and b, the constants of the fitting
curve in terms of the moments of the observed distribution, but the
results may be rendered more concise by WTiting

^i=fiyH'% ^■i=H'i'H-'2' ' • • (10)

whence

p — (2^2-3^i-6);2(o^2-6^i-9). .... (H)
g=-6 = -V(i^-A)-(i82+3)/2(5^2-6^i-9), . . (12)
r = -/i2(4i82-3j8,)/2(5^2-6i8,-9) .... (13)

And K, the criterion for fixing the type of curve suitable to the
statistics given, is immediately deduced from
K=q-/4:2)r
=^,{^,+Sr/4{4^,-3^,){2^,-S^,-6) . . . (14)

Also, since — vanishes when x = — b, this fixes the mode relative
dx

to the origin. But the origin is now at the mean, so that
mode-mean=-6 = - V(/^2i8i) • ()S2+3)/2(5j82-6j8i-9) (15)

And

skewness = (mean— mode)/S.D.

= 6/V(M2)

=Vi3i(i82+3)/2(5iS2-6^,-9) . . . (16)

CHAPTER XVII

APPLICATIONS OF CURVE FITTIJfG

We are in a position now to test the application of these principles
to given frequency distributions and we shall start by trying to
find a curve to fit the record of marks obtained by 514 candidates
in a certain examination (see p. 25).

Example (1). — This example is chosen because it turns out,
when we come to evaluate k, that it is well fitted bj' the normal
curve, Type VII, which is one of the simplest and at the same time
the most important of all the tyipes discussed. Before we start
the numerical part of the work it will be well to express the
constants y^ and a of this curve in terms of the moments of the
distribution.

The equation of the normal curve is

X-

y=yoe"2^'.

If N be the total frequencj^ wc have by equation (4) bis, p. 202,

r-fco
N=l ydx

J -co

^Voj e-='''-''-dx.

J -co

dx
Put x-l2a-=^^-. so that — =ct\/2 and when a:=00, |=00 also.

Thus N=?/oa\/2 e-'V|

.'-co

z=y(^aV2V7T (see Appendix, Note 8)

= V(277)c72/o . . . (1)

?06

Again

APPLICATIONS OF CURVE FITTING

/■+00 / r + ca

/^2— / yx-dx \ ydx

J-(X> I J-oo

207

2j/„

"2"'

N

since

vanishes at both limits.

Therefore, yL^ = '\/'2 . ay^VTr . cr-/N=cT-, by (1).

In fact, a is simply the S.D. of the distribution.

And yo=N/\/(27r) . a.

Table (41). Distribution of Marks obtained by 514 Candi-
dates IN A CERTAIN EXAMINATION.

Mean No.

of

Marks.

Deviation
from 33.

Frequency

of
Candidates.

First
Moment.

Second
ISIoment.

Third Fourth
Moment. Moment.

3

8
13
18
23
28
33
38
43
48
53
58
63

{X)

-6
— 5
-4
-3
_2
-1

+'l
+ 2
+ 3
+4
+ 5
+6

(/)

5

9

28

49

58

82

87

79

50

37

21

6

3

■(fa)

- 30

- 45
-112
-147
-116

- 82

+ 79
+ 100
+ 111

+ 84
+ 30
+ 18

(fa')

180

225

448

441

232

82

79
200
333
336
150
108

(fa')
-1080
-1125
-1792
-1323

- 464

- 82

+ "'79
+ 400
+ 999
+ 1344
+ 750
+ 648

(fa*)
6480
5625
7168
3969
928
82

79
800
2997
5376
3750
3888

—

—

514

-110

2814

-1646

41,142

208 STATISTICS

The first 4 moments referred to 33 as origin and with the class
interval, 5 marks, as unit of deviation, are

-110/514, 2814/514, -1646/514, 41142/514.
The arithmetic mean of the distribution
=33+5:k
=33+5(-ii|)
=33-5(0-214008)
=31-92996.

The second, third, and fourth moments referred to the mean as
origin, and retaining five marks as unit of deviation, are given
(see Appendix, Note 5) by

7;2=2814/514-.t2_5-42891

1,3=- 1646/514- 30=1-2-^3 =,0-29296

After making Sheppard's adjustments

these become

/X2=5-34558, /z3=0-29296, )U4-76-11436.

Thus iSi^/x-a/jiASg =0-00056, ^2=/^4//>t-2=2-66365.

Hence «=^i(^2+3)V4(4i32-3^i)(2^2-3j8i-6)

=(0-00056)(5-66365)2/4(10-65292)(-0-67438)

= -0-00063.

Since k and jS^ are small and ^2 ^o^s not differ greatly from 3, making
p and q small, we may fit a normal curve to this distribution.
The appropriate normal curve is

where ct-=/X2 =5-34558 (5 marks as unit),

2/o=N/V(277/x2)=514/\/277^(5-34558)^=88-6903.

Hence the required curve has for its equation, writing results to
three significant figures.

Now the mean of the distribution is at 31-92996, where the
central ordinate of the normal curve is erected, and the distance
of any x, say ^33, from this point

= (33— 31-92996)/5 (expressed with 5 marks as unit)
=-0-214008.

APPLICATIONS OF CURVE FITTING 20d

Any other x may be found in the same way and y can then be
deduced from the equation of the curve by taking logs, thus

log.,,=log„88.6903-^-^^^log„e

=1-9478762- (0-0406218)x-.

This enables us to calculate the ordinates of the normal curve and
thence we could evaluate the areas by successive applications of a
suitable quadrature formula.

We can, however, get the areas direct by using a table of the
probability integral, such as that due to Dr. W. F. Sheppard (see
pp. 284, 285). In that case the corresponding abscissae have first
to be expressed in terms of the standard deviation as unit, e.g.

a;4Q.5=40-5-31-92996=8-o7004,
and CT=5V'(5-34558) = 11-56025,

where the factor 5 is introduced because 5 marks was the unit in
the calculation of /it, (a process equivalent in effect to that previously
adopted).

Thus a;4o.5/a =0-741336
=i, say.

The area of the normal curve up to the abscissa xja or ^

= 1 ydx

.'-co

J ~OD

■'-"°V277

=n/^ zd

J -ca

=N . A(l+a),

1
where - represents the area of the curve z= — =e~^^'^ between

2 V277

0 and ^.

210

STATISTICS

Sheppard's Tables give the values of |(l+a) for different values
of ^, and \\ hen

^=0-74, i(l+a) =0-7703500
^=0-75, 1(1 +a) =0-7733726.

Therefore, by interpolation, when

^=0-741336, i(l+a)=0-7707538.

Thus the frequency of candidates with marks lying between 0 and
40-5

=514(0-7707538) =396-17.

Similarly the frequency of candidates with marks lying between
0 and 45-5 =452-20.

gyOy - ~r " - - ...

? Li-f--i

5 . . /Jr:-5i _ .. _ _ _

'S^ . ._ - ,? !-- .: - -

6 o?r ± . ^.j.-t. .\ - X It _

" I^fi .,- -^ -.

^ X 2 ^ *-S-- . - . ..

" S" ' i 1 ^t- ---

" " ft" , .... I -, . _ . - .

±"'S" z/_. s. _ .--_.

' Sfin i/ -.S ...

z'S^i'S-'. :. - ±.-., -- ^ --

5^ ,7 . , . r . _ ..

._. __ f. i j^^H

is£ _ .- :i : :_- - .-a

" -><§ _ ._ ,1 __ .__ _; .. _

_ ^ _. _- — J ^ ^. - - --

is - --W ._ - .. - ^ . - -. .

" ±u i^ - i

ia In ,' - -^. - ---

""ig' " \ " — :::::"": i — " "" : :: : :

ijSr : : ■ __- ->.----

±s: .^ I _ _ _ _ __ L . _ . ..

iS Z _ - -.. - : , .

{§ ^ _.----

- -St""' z " ' "--' ' _ ^:":""::~ :

"

" j? "" -- j'l z ' :;:. _;::: : : :+: ::

" ^5:" ---{'- : i: : ;:: :::: : : .:_5;:: :

;;

feiJIIIl-lllllllllllllllllllllNlllllllllllllllllllllllllllllllllflWWIJ

20 30 40

Marks obtained
Fio. (35).

50

60

70

Hence the normal frequency for the group with 43 as mean
number of marks =56-0, and the same method gives the area for
any other group.

The histogram of the observations and the curve plotted from the
ordinates are shoA\Ti together in fig. (35).

In Table (42) are set out the calculated normal frequency (col. (4))
for each group alongside the corresponding observed frequency
(col. (2)), and the differences between the two are shoMH in col. (5).
We want to know whether the fit is a good one.

APPLICATIONS OF CURVE FITTING

211

Table (42) Comparison of Observed and Normal
Frequencies in Examination Example.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Meau No.

Normal Frequeiic}-.

Ratio of No.

of
Marks.

Observed

Deviation.

Sij. of

in Col. (6) to
No. in Col. (4).

Frequency.

Ordinates.

Areas.

Deviation.

3

5

3-9

5-7

+ 0-7

0-49

0-09

8

9

10-4

10-7

+ 1-7

2-89

0-27

13

28

23-2

23-5

-4-5

20-25

0-86

18

49

42-9

43-1

-5-9

34-81

0-81

23

58

65-8

65-6

+ 7-6

57-76

0-88

28

82

83-7

83-1

+ 11

1-21

001

33

87

88-3

87-6

+ 0-6

0-36

0-00

38

79

77-3

76-8

— 2-2

4-84

006

43

50

561

56-0

+ 6-0

3600

0-64

48

37

33-7

34-0

-3-0

900

0-26

53

21

16-8

17-1

-3-9

15-21

0-80

58

6

7-0

7-2

+ 1-2

1-44

0-20

63

3

2-4

3-5

+ 0-5

0-25

0-07

••

514

511-5

513-9

••

184-51

X^ = 504

Now with this object we might square each difference as in
col. (6), sum the squares, and find the mean square deviation by
dividing by the total frequency ; this, after extracting the square
root, would give what might be called the root-mean-square error,
regarding the theoretical values as the true ones. In the above
example it

=V(184-51/514)=0-599.

But this form of result, while it may be useful in some cases,
e,.g. in comparing two distributions of the same kind to some
theoretical series, is open to objection ; for one thing it treats all
the differences as if they Avere of equal importance in absolute
magnitude, but a difference of 2, say, in a normal frequency of 10
is clearly more serious than a like difference in a frequency of 60.
The objection, however, goes deeper than that ; even when the
root-mean-square deviation is found we are at a loss to estimate
its precise relationship to the quality of fit, as there seems to be no
definite connection between one distribution and another of a
different kind : there is no standard case, so to speak, to which we
can always appeal, where the fit is agreed to be good and supplying
therefore a suitable root-mean-square deviation for comparison.

w-v-^'"^- "v.^-^-Tf e-- ;

(^^-r

212 STATISTICS

This leads us to the question : What constitutes goodness of
fit ? Suppose by some means we have selected a theoretical or
empirical formula to describe a certain frequency distribution in a
given population ; if the frequency values observed do not diflfer
from the theoretical frequencies by more than the deviations we
might expect owing to random sampling, then clearly the fit may be
regarded as a good one. And we have a measure of the fit if we
can find the proportion of random samples, of the same size as the
given distribution, showdng greater deviations from the distribu-
tion given by theory than those which are actually observed.

Now Professor Karl Pearson has shown how this proportion can
be calculated \Pliil. Mag., vol. 1., pp. 157-175 (1900)] ; he finds the
probability that a random sample should give a frequency distribu-
tion differing from that Avhich theory proposes by as much as or by
more than the distribution actually observed. This probability, P,
is a function of ^, where

y and y' representing the theoretical and observed frequencies for
any particular group and the summation is to include all groups.
It will be noted that this expression gives each difference {y—y')
its appropriate importance by relating it to the frequency y of its
own group.

A table in Biometrika (vol. i., pp. 155 et seq.) gives the values of P
corresponding to different values of y^ (including all integral values
from 1 to 30) and to values of n' , the total number of frequency
groups, from 3 to 30. (see also p. 285). The mathematics in-
volved in finding P is difficult, and the reader who wishes to enter
into it must consult the original memoir, but the utility of the
function has been proved by experience and it is readily applied
in a particular case.^^<,^£:i-^*'^''^S

In the above example p^^ is found from col. (7) : it equals 5-04,
and from the table of values of P, when n' =13, we have

P =0-957979 when ^^=5,
and P=0-916082 when x'=^-

Therefore, by proportional interpolation, when ^-=5-04,
P=0-956303. Thus, supposing our data to follow the normal curve,
in 956 random samples out of 1000 we should expect to get a
worse-fitting distribution than that given by the sample actually
observed. We may therefore conclude without hesitation that
the normal curve provides an excellent fit in this particular instance.

APPLICATIONS OF CURVE FITTING

213

We pass on now to fresh distributions to illustrate some of the
other types of frequency curve.

Example (2) deals with the percentage of trade union members
unemployed at the end of each month for the years 1898 to 1912
[data from the Sixteenth Abstract of Labour Statistics of the United
Kingdom, Cd. 7131]. Table (43) shows the distribution of the
180 records according to the percentage unemployed.

The deviations are measured from the centre of the group (3-9— 5-2)
as origin, and the class interval (1-3 per cent.) is taken as unit of
deviation as usual.

The first four moments are : —

-29/180(=a-), 425/180, 397/180, 3053/180 ;
i.e. -OlGllUl, 2-3611111, 2-2055556, 16-9611111.

Table (43). Distribution of Unemployed Percentages
OF Trade Union Members

Percentage

Devia-

Fre-

First

Second

Third

Fourth

Unemployed.

tion.

quency.

Moment.

Moment.

Moment.

Moment.

0—

-3

0

0

0

0

0

1-3—

_2

33

-66

132

-264

528

2-6—

-1

57

-57

57

- 57

57

3-9—

, ,

41

, ,

. .

5-2—

+ 1

24

+ 24

24

+ 24

24

6-5—

+ 2

10

+ 20

40

+ 80

160

7-8—

+ 3

11

+ 33

99

+ 297

891

9-1—

+ 4

3

+ 12

48

+ 192

768

10-4—

+ 5

1

+ 5

25

+ 125

625

••

••

180

-29

425

+ 397

3053

Referred to the mean,

4-55 -fl-3.€- =4-3405556,

the second, third, and fourth moments are (see Appendix, Note 5),
,;2=2-3611111-.f2=2-3351543,
r3=2-2055556-3:ri/2-:c3=3-338395,
r4=16-9611111-4i;-;'3-6:cV.2-.f*=18-74817.

Owing to the very doubtful contact at the beginning of the curve
Sheppard's adjustments were not made in this case, but the rough
moments as calculated above were used.

214 STATISTICS

Thus jSi =7.2 jj,3^ ^0-875242

iS2=/V'^'2=3-43817
and /.=iSi(|8.3+3)V4(4^o-3^i)(2^2-3^,-6) = -0-466.

Since k is negative the fitting curve should be of Type /.,the equation
of which is

y=yo 1+-

where mja-^=7n2la.2, and (ai-l-ao)=6, say.

It is therefore necessary before going further to determine y^, a^,
tto, b, m^ and m^ in terms of v^, v^, v^, or jS^ and jSa, the constants of
the distribution.

The value of y^ is found to be most conveniently expressed as a
Gamma function which is defined, with the usual notation, thus : —

r{n) = j x'^-'^e-Hx,

whence it follows that T{k-^\)=kT{k). [See Appendix, Note 9,
also p. 285.]
Also, if

B(m, n) = \ .r™-! {l — xy-'^dx

it may be easily shown that

B(m, 7i)=r{m)rin)/r{m+n). [See Appendix, Note 9.]

The general method of procedure in determining the constants
for all the different types is : —

1. Express the fact that the area of the curve is a measure of

the total frequency of the distribution — this enables us to
find ?/o.

2. Find the nth moment of the curve with regard to some fixed

■origin— giving ?t particular values, 1, 2, 3, 4, this leads to
the determination of jXo, /Xg, jx^, jS^, jSg in terms of the con-
stants of the curve, and thence to formulae for calculating
the constants.

Once found, the same formulae may be used, of course, in all
cases of the same type : we have only to replace letters by the
numbers for which they stand.

Applying this method to the Tyj)e I. curve, we have

[ + "■2

N= ydx

J -a-i

i-ai

I- + 02

a.^tto

^" ' {a.i-xY'Ha^-xY'^dx

APPLICATIONS OF CURVE PITTING 215

Put {a^-\-z) = {ai-\-a.^)z, so that {a2—x) — {ai-\ a2){l—z) and

-J- = (aj+a^'>)=6 ; therefore
dz

a^"'>a.^"'-^ Jo

(2)

.l"'^W2./'2

BCm,+ l,W2+l).

m{ 'm.2

Hence yo= . --^ — ~ — — •

b (m,4-m2r+"'^ r(m,+l)r(m2+l)

Again, %'„=/ ?/(ai+a:)V^

• -"i

is the nth moment of the distribution referred to (— a^, 0), the
point where the curve starts from the axis on the left-hand side,
as origin.

Therefore, as above,

= 6"Nf 2"'i + "(l-2)'"2C?2// 2"'i(l-2)™2c?2, by (2).

.'o / .'o

Hence,
^'"=6"r(Wi+w+l)r(mi+m2+2)/r(?Wi+l)rK+?W2+w+2)

= 6"(mi+n)(mi+w— 1) . . , (mi4-l)/(^^i+^2+^+l)(^i+*^2+^)
. . . (mi+m2+2),
by repeated application of the relation r{k-\-l)=kr{k).

Putting w=l, 2, 3, 4 in succession, we have
^'j=6(mi+l)/(mi+m2+2),
/Lt'2=6^(mi+2)K+l)/(mi+m2+3)K+m2+2),
^'3-63(mi+3)K+2)(mi+l)/(mi+m2+4)(mi+m2+3)K+m2+2),
/^=6*(mi+4)(mi+3)(mi+2)(mi+l)/K+m2+5)(mi+m2+4)
(mi+?W2+3)(mi+rrt2+2).
These relations are rendered more concise if we ^^Tite
7ni-]-l—m\, 7712+1 =w'2, 7Wi+W2+2=r;
thus iJL.\=b'm\/r

fji\^b-m\{m\+l)lr{r-^l)

fi\=b^m\{m\+l){m\-\-2)lr{r+l){r+2)

^'^=6W^(w'^+l)(m'i+2)(w'i+3)Mr+l)(/-+2)(r+3).

216 STATISTICS

To get the corresponding moments referred to the mean as
origin we have the relations : —

which, after some straightforward reduction, give
H2=b-m\m' Jr-{r+l)
fjL^=2b^m' I'm' ^i^n' 2—m' i)/r^{r-^l){r-ir2)
^^=3/;Wim'2[m\w'2(r-6)-f2r2]/r*(r+l)(r+2)(r+3).

Thus i3,=^^//x32^^*'^"^^^"^^^'^-^'^)' /b^m'\m'\

=4(m'2-m'i)2(r+l)/m'iw'2(r+2)-
=4:{r^-47n\m' .2)ir+l)/m\m' r,{r+2)\

Therefore, _Il_ =M±?1V4 . . . (3)
m\m'2 4(r+l)

^ . ^ , , Sb*m\m' Jm\m' Jr-Q)-\-2r-] /b^'\m'K
Agam, iSo =fj-i iJi-o =- ^^ — - — ^^ — — / —

3[m\m\{r-6)-\-2f-] (r+l)

m\m\ {r+2)(r+3)

Therefore. J!% =.-r-,6+^^'^^^ ... (4)
w'im'2 3(r+l)

Combining ,3) and ,4). '^^+S=0-r+^^^^^,
4(r+l) 3(r+l)

whence r=608.-i3,-l)/(3^,-2j3,+6) . . (5)

Again, since /x2=6-m'im'2/r-(r+l),

therefore 62^/^2(^+1) • [i3i(r+2)2+16(r+l)]/4(r+l), by (3),

i.e. b^^V/I^A [iS,(r+2r+16(r+l)] .• . (6)

And m',w'2=4/-2(r+l)/[^i(r+2)- + 16(r+l)],

while m\-]-m'2=r ; hence w'^ and m'2 are roots oi

4r2(r+l)

WI-— rm+ - =0,

^,{r+2)'+ie(r+l)

r lY 167--(r+l)

the solution of which quadratic is - ± ^ / --

j8,(r+2)^+16(r+

.}

APPLICATIONS OF CURVE FITTING

therefore, m^ and m^* are respectively er[ual to

_2+ r(r+2)Vj3,

and ttj and a.^ follow from

a, a, b

217

^ 1 .

m.

m^ nil + nig

(7)

(8)

Applying these formulae to the ' unemployed ' example, ^we find

r=5-36048.
6=9-33236.

=0-169185.

m2=3-191295.

a2=8-86252.

a^ =0-469842.
Also ?/o=58-1282, and the equation of the curve is therefore

y = 58-l 1

0-470

X

8-86

The position of the origin, M'hich is at the mode, is given by

{mean-^node)=/z'i— ttj

_bm\ brrii

_^/m'i_m\-l\
V r r-2 /

=6

m

m

r(r-2)

(9)

thus,

r+2

mode =4-3405556- 1 . ^ ,

i^2 '■—2
in this particular case,

=2-3052009.

[* When Mj is positive m.2 goes with the positive root of the quadratic, and
vice versa.]

218

STATISTICS

This enables us to Avrite do\\ai any x, and thence y by substituting
for X in the equation of the curve, which, by taking logs, may be
written

log y=\og Vfi+m^ log l + _ j+mg log ( 1-

e.g. for the x of the group (2-6— 3-9), bearing in mind that 1-3 is the
unit of measurement for x, we have

a:3.25=(3-25-2-3052009)/l-3=0-9447991/l-3.

Hence ( 1+' ^^^ ) =2-546835 ;

^^M =0-9179953:

a, ,/

m, log ( l+'^:25 )_o-0686892 ; w^ log 1-'^:^ =-0-118587 ;

\ «! ' \ «2 / .

so that log 1/= 1-7 14489,
and 2/3-25=^1 '^2.

Similarly the ordinates at the centre points of the other groups
may be calculated, but it must be remembered that the resulting
values are only a first approximation to the observed frequencies,
and a better series is obtained if, by using some good quadrature
formula, we calculate the areas for the successive groups between
the curve, the bounding ordinates, and the axis of x. Indeed in
the case of the group (1-3— 2-6) it is essential to do this, because
(1) the rise of the curve is so very abrupt as to render the deter-
mination of the single ordinate at the centre quite inadequate for
an accurate measure of the frequency in that group, and (2) a
portion of the group falls outside the range of the curve which only
starts at 1-6944063 (i.e. mode— l-3ai), and this has to be allowed
for in finding the frequency as represented by the area between the
curve and axis.

The base of the required area, range (1-6944063 to 2-6), was
therefore divided into eight equal joarts and the ordinates at the
points of division were determmed. The area was then found by
using Simpson's well-known formula : —

Area=iA[(t/o+2/op)+2(?/2+?/4+ . . . +2/2p-o)+4(?/i+?/3+ . . . +y2;,-i)]>

where h denotes the length of one of the equal parts into which
the base is divided and 2^ is their number ; in our case ^=4 and
/i=:l the class interval being the unit, and the result is to be
reduced in the ratio

0-9055937 : 1-3

APPLICATIONS OF f'lTRVE FITTING

219

in order to allow for the smaller range of this group ; we thus get
as the area for the group

-^^^^xJr(yo+y8) + 2(y2+v/4+y6) + 4(2/i+Z/3+y5+y7)]=37-39.
1-3 24

The observed and calculated frequencies for the whole series are
compared in Table (44), the remaining areas in col. (4) being calcu-
lated by the simpler but somewhat less accurate form of Simpson's
formula, when only three ordinates are used, namely,

,■+1

I yr/.r=i(i/_i + 4//(,+2/i).

Table (44). Comparison of Observed and Theoretical
Frequencies of Unemployed Percentages

(1) (2) (3) (4) (5) (6) (7)

Percentage
L' nam ployed.

Observed
Frequency.

Theoretical
Oidinates.

Frequency.

Deviation.

Areas.

Square of
Deviation.

Ratio of No.
ill Col. (fi) to
No.inC..l.(4).

1-3—
2-6—
3-9—
5-2—
6-5—
7-8-
9-1-
10-4-

33
57
41
24
10
11
3
1

55-3*

51-8

37-8

24-9

14-8

7-7

3-3

1-0

37-4

51-6

37-8

25-0

14-9

7-8

3-4

1-2

+4-4

-5-4
-3-2

+ 10

+ 4-9

-3-2

+0-4

+0-2

19-36

29-16

10-24

1-00

24-01

10-24

0-16

0-04

0-52
0-57
0-27
004
1-61
1-31
0-0.5
003

••

180

••

179-1

x2=4-40

To test the goodness of fit we have n''=8, ^-=4-40, whence, by
means of the P table, P=0-731852. Thus, roughly, Ave may say that
three out of every four random samples of 180 records would give a
worse fit with the proposed curve than is given by the actual distribu-
tion observed, so that the fit may be regarded as quite a reasonably
good one. This conclusion is also supported by an examination of
the curve which has been drawn, fig. (36), with the histogram of
the given statistics.

Example (3). — The data for this example concerning infectious
diseases will be found in Table (16), p. 62 (or, see p. 224) ; the
reader should work out the moments for himself and verify the
folloM'ing results : —

[* The ordinate in this case cannot be accepted as an approximation to the
frequency given by the curve. ]

220

STATISTICS

The first four moments referred to 7 as origin are

0-282158, 4-86307, 17-4855, 129-394.
Referred to the mean, 7-564316, the three latter become
1/2=4-78346, 1/3 = 13-4140, j/4=lll-964.

// we do not assume high contact at the terminals, and certainly at
the lower end it is doubtful, we deduce from the above values of
the moments that

^1 = 1-64396, ^2=4-89321, /c = -l-53.

Thus the fitting curve is of Tyjpe I. and its constants, when calcu-
lated, are

y=ll-7819. mi=0-31171. ^2=9-47020.
aj =0-79216. ^2=24-0671. 2/o=60-363.

ou

--

_

z

!r_

-^

_,

_-

_l

r-h-

hi^H

^

pl

--

-

U

^

u

U-

--

-,

_

_

-.

^-

-

—

y

'

^

-

"1

^

~

"

"

"

"

"

~

■^

~'

~~

"

~

~

"

H

+

50

\

^

^^

1

s,

o

_

-

-.

y-

\

M

-.

J

L

-

-

-

^

_L

-

J

-

__

!^40

-

^

--T

T

-

-

-

-

--

-

-'

r

n

-^

-

--

=>30

1

:

::

_

-J

.

__

_

_

„

_

^

-

_

__

_^

^

-

-

-■

t

-

-

1

--

--

-

- H

-

-J

^

-

-

--

--

^

-

-

--

3i

1

o

\

c

'

'

N

2^20

'

'

1

■

'

"s

— i

'

■«

■i.0

_

^

_

._

_ ^

|,

_

__

^

_

__

.

+

_

J!

s

-

_

__

__

J

_

_

-_

-

■-

--

■^

-

-

■-^

T

-

■-

-

'■-H

^

v^

-

-

--

■■-

-

-

-

--

~~l

*.

.

J

-

:

:i

= -

L

-'-

:

-

i.

3
n)

~.z

-;

;-

:

-_z

-

_

-

J

-

"

? =

s

^

s

=

f

Y.

0 1 2g3 425 6 7 8 9 10 11 12

Percentage Unemployed
Fig. (36).

The equation of the curve is therefore, retaining three significant
figures throughout

y=60-4 1+-^ 1-^ •

^ 0-792/ \ 24-1,/

The curve starts at 2-02904 (so that the first group of observations
lies wholly outside its range) and ends at 51-7475. It is drawn,
together with the corresponding Mstogram, in fig. (37).

Supposing, just for the sake of comparison, we assume high
contact at the terminals and attempt to fit the given distribution
with a Type III. curve, to which Type I. is closely related.

We then have, after making Sheppard's adjustments,
/X2=4-70013, /X3=13-4140, /X4=109-601,
whence ^i=l-73295, ^82=4-96129, /c = -l-47.

It will be noted that the theoretically correct type to take here
again is Type I., but this was discarded because, when attempted,

APPLICATIONS OF CURVE FITTING

221

it led to a curve starting at a point corresponding to a disease rate
of 3-385, so that the central ordinates of each of the first two
observed groups lay outside the curve altogether.

Type III. curve is of the form

y

70

60

■^50

a 40

to 20

tM

10

■ ■■ ■" " ■■ ■ -

::i::i!^-i::::::"::::i::i::::::i::i::::::::"g:i::::"::

----/::2::s:::::::i::::i::i::::::+:i:::::i:::i:i::i::::i:::

±-/-%, - \

^ 2 . \\ ~X

4 ^

/ \ V Tvnp I

:fc-: \\~ — TZ

, -A -^Type-Ill.

-::- :S;,:::i:ii::="=== -JiiiS::: : :::+:+:-:

_.._ __ L ^^ _ _ __ _ _

:::: ::::: ::::::-:!^r::=::==^======:=:==========:====:==:=:

___ "-^"^S 3: J i

:: i:._-t :: ^- :-35^:

i:i^:::±i::::::::::i::::^^_::i:::::::::::::::::i:::::::S:::

-— -+ 1 5; ± :::

]- — ^r :i 1 -N^; 1 '

1 Srf§ r§ TT*ti°+*44-U-l-L 1 1 1 1 1 1

10 15 20

Disease Rate per 1000 persons liu'mg
Frn. (37).

25

30

To express the constants in terms of the moments, noting that the
curve starts from x = — a on one side and goes ofif to infinity on the
other, we have

-co

N--/ ydx

J - a

_Jo I e-yx^a-]-xYdx (where ya=p)

a J-a

^y^-i e-y^ya+yxfdx
=J\ ev« /■ e - ^■^"+y'\ya-\-yxYdx

P J-a

=^^ I e'^z^'dz (where ya-i-yx=z)
yp^Jo

=''^€r(;>+l).

Therefore, y„=Np''+VaeT(p-|-l)

(10).

222 STATISTICS

Again, the nth moment of the distribution referred to (—a, 0)
as origin is

Therefore, by (10),

-1 /n

Hence,

=r(2?+w+i)/y"r(p+i).

li\=T{p+2)lyT{v+l)Mv+\)ly

/x'2=r(p+3)/y'^r(i>+l)={i>+2)(i>+l)//
)a'3=r(p+4)/y3r(i)+l) = (i>+3)(;)+2)(^+l)//.

Transferring to the mean as origin we have for the moments, since

/i2=/2-*^ = (P+l)/y'"
/^3=/^'3— 3^/i2— ^^=2(jj+l)/y3.

Hence, combining these last two equations,

y=2fijfi,, p^lW/^'a)-! • • . (11)

In our particular case these equations give

y=0-700780, 2^=1-30820, a=l-86678,

and, therefore, by (10),

yo=55-3323.

Hence the curve is

y=55-3e-°"Wl+ — ) .

V 1-87/

The equation of the curve, on taking logs, gives
log 2/=log y^—y log lo^ • x+p log ( 1+^

=l-742979-0-304345x+ 1-30820 log (l + x/1-86678).

=

-(i^+l)y-i'

'ly

=

'

mode =

=7-564316-

2-853960

A

•« —

Mode

Mean

APPLICATIONS OF CURVE FITTING 223

Before Ave can go on to calculate the ordinates of the curve we
need to know where the origin lies, and since it coincides with the
mode it may be found from

mean— mode =/x' ^ — a

. (12)
Thus, mode=7-564316-2-853960=4-71036.

Suppose now we wish to calculate the ordinate corresponding to
the X of the centre point of group (6—8), we have

a;7=i(7-4-71036)
=1-14482,

bearing in mind that the unit is a rate of 2 per 1000.
Hence, substituting this value in the equation for log y,

log 7/7 = 1-666278
?/7 =46-374,

and similarly any other y may be found.
The curve starts at

mode-a=4-71036-2(l-86678)=0-97680,

so that the range of the first group as determined from the curve is
(0-9768-2), and not (0—2) as in the observations.

The ordinates and afterwards the areas, calculated by a method
somewhat similar to that indicated in Example (2), were determined
for each separate group of observations, and the results for both
Tjrpe I. and Tjrpe III. curves are compared in Table (45).

Type III. curve is drawn on the same diagram, fig. (37), as Type I.
curve and the observation histogram, and the result lends emphasis
to an important jDoint, namely, the necessity for replacing ordinates
by areas to obtain the frequency proper to any group.

In order to get a measure of the goodness of fit in each case,
the function P was calculated, but in the Type I. comparison the
first group had to be omitted to avoid the infinite term which would
have resulted in v^^^, owing to this group falling right outside the
curve, that is to say, the test had to be confined to toA\Tis in which

224

STATISTICS

the observed ease rate was not less than 2. The values found for
P were : —

Type I.— P=0-34307,
Type III.— P=0-46298,

so that in every 100 samples containing 241 observations each, we
should get, roughly, 34 deviating from the Type I. curve and 46
deviating from the Type III. curve, at least as widely as the given
distribution. In neither case can the fit be regarded as a very
good one, but the failure is only marked in one or two groups, such
as that of maximum frequency, where there may be other than
random causes to account for it ; e.g. where isolation is inefficient
the disease is likely to spread, one case infects another : in other
words, the events are not independent.

Table (45). Comparison of Observed Distribution of In-
fectious Disease Rates, notified in 241 large Towns of
England and Wales, with Theoretical Distribution.

(1)

(2)

(3)

(*)

(6)

(6)

Observed
Frequenc}-.

Theoretical Frequency.

Case Rate.

(fi-f)Vfi.

{fs -/)■//,.

Tvpe I.

Type III.

0—

(/)
5

(/x)

if.)
6-6

0-39

2

39

52-6

43-7

3-52

0-51

4—

69

55-4

54-3

3-34

3-98

6—

41

43-2

46-2

Oil

0-59

8—

29

31-2

33-6

015

0-63

10—

22

21-5

22-4

0-01

001

12—

16

14-2

141

0-23

0-26

14—

7

91

8-6

0-48

0-30

16—

5

5-6

51

006

000

18—

3

3-3

2-9

003

000

20—

4

1-9

1-7

2-32

3-11

22—

0

10

0-9

1-00

0-90

24—

0

0-5

0-5

0-50

0-50

26—

1

0-3

0-3

1-63

1-63

••

241

239-8

240-9

X\ = 13-38

X% = 12-81

Example (4) refers to the wages of certain women tailors previ-
ously recorded in Table (11), p. 41. The data as given in the
original suffered a disadvantage common to such statistics : at

APPLICATIONS OF CURVE FITTING 225

either end the grouping differed from that in the centre, two or three
classes being lumped together owing to the smallness of frequency
in each. The figures ran thus : — Under 5s., 19 ; 5s. and under 6s.,
180 ; 6s. and under 7s., 384 ; ... ; 23s. and under 24s., 64 ;
24s. and under 25s., 54 ; 25s. and under 30s., 122 ; 30s. and over,
36. They were recast in the form sho\\Ti in Table (46), suggested
by an examination of the histogram, in order to make the fitting
simpler.

The first four moments calculated from this adapted table and
referred to 12s. as origin are : —

/i=0-556718, i;'2=5056373, ^^'3=16•70163, i.'4=123-7691.
When referred to the mean, 13-113436, the last three become

1/2=4-746438, i/3=8-60179, i/4=95-6914 ;
or, after making Sheppard's adjustments,

/^2=4-663105, /Lt3=8-60179, /m4=93-3474;
therefore, ^1=0-729713, /S2=4-29291, «=l-63.
The curve is thus of Type VI.,

y=yo(x-a)iVx'".

To calculate the constants, the nth moment about the origin is
given by

=2/o j{x—af'\v''-'''dx

•0 ,^1-2)"^ a-31 / 1\^ / ^ a

- — • — -(i{ — ^ )dzl where x=-

d'li-'.-z-n-l

Thus, putting n=0,

N=-^B(<7i-^.,-L (?3+l) . . . (13)

and ^'^=a-riq,-q,-l-n)r{q,)!r{q,-n)r{qi-q,-l);

therefore, fi\=ar{qi-q^-2)r{q,)/r{qi-l)r{q^-q.,-l}
=a(q,-l)/{q,-q,-2).

Also ^' Jix\_^=ar{qi-qi-l-n)r{qi-n+l)iriqi-n)Tiqr-qi-n)
=a{q^-n)/{qi-q2-n-l).

226 STATISTICS

^Blice fM',^aHq,-\){q,-2)l{q,-q,-^2){q,~q,-3)

IJi',=a^{q,-mh^2)iq,-3)l{q,-q,-^2){q,-q,~3)iq,-q,^A)

(?i-?2-4)(?i-<?2-5).

But these relations are precisely the same as those of Type I. with a

in place of b, —q^ in place of m^, and g, i^i place of mg, so that
(l+Qj)' (1 — Qi)* a-re the roots of

q2_j.q+4r2(j.^l)/^^^(j.+2)*+16(r+l)]-0 . . (14)

where T=6{^,-^,-l)l{e+Z^,-2fi,) .... (15)

Also yo=Na'i-''-^-T(qJ/r(Q,-q2-l)r(q,+l), by (13) . (16)
and a is given by

/x,= a^(l-q,)(l+q2)/r^(r+l), .... (17)

fjL^ being the second moment of the given distribution referred to
its mean as origin.

The distance of the mean from the origin is

yLi\=a(qi-l)/(qi-q2-2), .

and this fixes the origin, for the mean is known directly from the

statistics.

dy
To get the mode, use the equation of the curve, putting — =0

dx

and we have

origin =mode — ag'i /(g'l — g'j) •

Combining this with

origin=mean— a(g'i— l)/((/i— 92— 2)
we have

mean-mode=a(qi+qj)/(qi-q2)(qi-q2-2) . . (18)

Applying these formulae to the case of the women tailors,
r=-38-7698, 9i=51-5269, q2 = 10-loll, a=2M1018,

and the equation of the curve is

y=yo(x-21-l)"Vx"^

where log ?/o = 68 • 8254 .

Also the origin is at —41-9104, the mode at 11-4498, and the maxi-
mum theoretical frequency is 2299.

[* Wlien fjL.^ is positive (l+q^) goes with the positive root of the quadratic, and
vice versa.]

APPLICATIONS OF CURVE FITTING

227

Table (46). Distribution of Wages of certain
Women Tailors, Actual and Theoretical.

Wages.

Frequency.

Wages.

Frequency.

Actual.

Theoretical.

Actual.

Theoretical.

Is.—

3s.—

5s.—

7s. —

9s.—

lis.—

13s.—

15s.—

17s.—

5

14

564

1243

2045

2339

1815

1432

854

1

52
452
1332
2096
2255
1898
1353
859

19s.—
21s.—
23s.—
25s.—
27s.—
29s.—
31s.—
33s.—

523

262

118

64

43

27

15

9

503

278

147

75

38

19

9

5

••

••

••

11,372

11,372

The theoretical and actual frequencies are compared in Table (46)
and the curve is drawn with the histogram in fig. (38).

2500

2000

S

O1500

3 1000

500

^Et!:

cy-

5/-

,0/-

15/- 20/-

Rate of Wages
FV». (38).

25/-

30/-

35J,

Example (5) discusses the distribution of frequencies of specimens
of Anemone nemorosa with different numbers of sepals, recorded by
G. U. Yule [Biometrika, vol. i., p. 307).

228 STATISTICS

The first four moments referred to 6 as origin are

i,'^=0-508, i/'2=l-012, z^'3=2-476, v\=9l24:.
Referred to the mean, 6-508, the last three become

1/2=0-7539360, 1.3=1-195905, i/4=5-459941.

The contact, at one extremity certainly, being doubtful, Sheppard's
adjustments were not made in this case. Hence,

jS^=3-337259, /32=9-605476, «=l-46.

Since k does not differ greatly from unity an attempt was made to
fit the observations with a Type V. curve, namely^
y=yoX-''e-^^
The nth moment about the origin is given by

. ^fx\= jyx^dx
Jo

(since, p and y being positive, y vanishes at a;=0 and at cr=oo)

=2/oy " ~ ^' "^ M z^'~'^~~e~~dz (where z — yjx)
Jo

=7j,y--^+'r{p-n-l).
Thus ^=yoy-P+^r(p-l).

And fi' Jfi' n-i=y/{p-n-l).
Hence ft'i=y/(2)— 2)

fi'o_=yV{p-2){p-S)

/x'3=y7(P-2)(i?-3)(2)-4).
Referred to the mean as origin, the last two moments become

fM,=4yyip-2np-Z)ip-4),
whence

P,=fJiMfx\=lQip-S)/{p-4r^=[lQ{p-^)+lQ]l{p-^r-;

this gives a quadratic for (^—4), one solution of which is

p-4=[8+4V(4+A)]/iSi. • • • (19)

the positive root being taken in order to get a real y.

Thus y*=(P-2)V[(P-3W • • . (20)

and y„=Ny''-Vr(p-l) .... (21)

Since fji\=y/{p—2), the position of the origin is given by

Origin = Mean -y/(p- 2) . . . (22)

ALso the distance of the mode from the origin is y/p, so that all
the constants of the curve are readily determined.

[* The sign of y is taken to be the same as that of M3.]

APPLICATIONS OF CURVE FITTING

229

In our particular case, we get

P-9-G43840, y =17-10758,
and the curve is

600

500

Co

400

300

5^200

100

y=yoX-'"e

9 64p-171/X

~]

"

■

~

Ji

\

I

J

f

t:

x

_r

r

4

^

l1

_n

I

T -4

- t

r- -

■t

-X -

t

I ■ ^

_I-

L

4-

^ ^

'4"

1

7

- -h -1

I

_ I

_t

. i

t

' 4 -1

t

Jt^

- -X

T

1 J

i

4

1-

1

1

I

t

1

{

I

1

\

t

\

\

\

*

\

\

'

1

—

1

y

\

J.

\

\

tt

t

\,

\

t

\

t "

s

\

.

s.

S

^

V

s

h^

T3 rt

K.,

.

1

7

O W

\ii

a

-xJ

— .>c-=iL___

^ r ^

1 "

::=

Oi

s

d

w

kxi

; „

^

Number of Sepals

Fid. (I^O).

10

12

where log i/o=9-38179. The origin is at 4-27 and the mode at 6-04.
The greatest frequency is 620 approximately, and the frequency dis-
tribution, calculating areas for the several groups as if they ranged
between (4-5— 5-5), (o-5 — 6-5), etc., is shoMTi alongside the observed

230

STATISTICS

distribution in Table (47). The curve is plotted in fig. (39) from the
ordinates which were calculated at the centre and extremities of
each group so as to enable Simpson's simple quadrature formula
to be used to get the areas.

Table (47). Distribution of Sepals of Anemone
Nemorosa, observed and calculated.

[Examples have been given above of five out of the seven different types
of frequency curve that have been enumerated. For further examples of
.all the types and a complete account of the method reference should be
made to Professor Pearson^ s memoirs, especially the following : —

Roy. Soc. Phil. Tram., vol. 186A/pp. 343-414 (1895), On Skew Variation
in Homogeneous Material ; and a Supplementary Memoir in vol. 197a, pp. 443-
459 (1901).

Biometrika, vol. i., pp. 265 el seq., On the Systematic Fitting of Curves to
Observations and Measurements, continued in vol. ii., pp. 1-23. Also vol. iv.,
pp. 169-212, which discusses various historical hypotheses made to generaHze
the Gaussian Law, the basis of the symmetrical normal curve.

A large number of highly interesting practical illustrations of Pearsonian
curve fitting occur throughout the pages of Biometrika, while W. P. Elderton's
Frequeiicy Curves and Correlation contains an admirably concise treatment of
the theory, with applications to meet more particularly the actuarial point
of view.

It should be stated that rival curves and methods have been proposed as
suitable for fitting certain t%-pes of frequency distribution, some of which have
scarcely received the attention and the trial they deserve. Among the most
interesting are those developed by Professor Edgeworth ; for some account of
his voluminous work upon the subject the reader may refer to several memoirs
in the Journal of the Royal Statistical Society, beginning December 1898
(the Method of Translation), among which the following are important as
giving more recent results of his researches : — ■

Vol. Ixix. (1906), The Generalized Law of Error or Law of Great Numbers.

Vol. Ixxvii. (1914), On the Use of Analytical Geometry to Represent Certain
Kinds of Statistics.

Vol. ixxix. (1916), On the Mathematical Representations of Statistical Data;
continued in vol. Ixxx. (1917).

Two memoirs may be cited as of particular interest — those of May 1917
and March 1918 — because they reply to criticism and draw a comparison from
their author's point of view between his curves and those of Professor Pearson.]

CHAPTER XVIII

THE NORMAL CURVE OF ERROR

Let us return for a moment to the general statement on p. 143,
that ' \^henevcr we have n similar but indej)cndent events happen-
ing in which the probability of success for each is j^, the different
resulting possibilities as to success are given by the successive
terms in («+/)", namely,

1 . ^

and their correspondent probabilities by the successive terms in
ip-\-q)", namely,

J. • ^

When we come to try and ajiply this theory directly to cases
other than those of random sampling in artificial experiments ^\ith
coins, dice, etc., we are faced at once with difficulties because of
the limiting character of the assumption on which the theory rests,
namely, that all the events are to be similar and independent. The
similarity demanded is of the same radical type as that existing
when we throw the same die or spin the same coin tAvice running,
and the test for it is that p, the chance of success, is to be the same
for every individual event. The independence is to be such that
no single event and no combination of events is to have any influence
upon any of the rest.

Now for most classes of events it is impossible to assign any
a priori value to p at all, still less can we be sure that p does not
change from one event to the next. For example, the chance of
death for soldiers in Avar-time varies from regiment to regiment
according to where they happen to be located ; for the same regi-
ment it varies from battalion to battalion according to whether
they are in tke trenches or behind the lines ; and from individual
to individual according to innumerable little accidents of time, place.

232

STATISTICS

and condition. Also, where the shells burst thickest, p increases
for any soldier there, but it increases also for his neighbour. Thus
the events in such a case are not similar, neither are they inde-
pendent.

Moreover, as it stands, the theory cannot be applied to any
distribution in which the character observed is capable of continu-
ous variation. Tliis difficulty, however, has been overcome, as we
have seen, by replacing the histogram representative of the binomial
by a continuous curve which at the same time serves to describe
the discontinuous series to a high degree of accuracy.

To illustrate how close this
description can be, even when 7i
is comparatively small, we will
fit with its appropriate normal
curve the symmetrical binomial
polygon formed by joining up
the summits of the ordinates
representing successive terms of
the series

erected at unit distance apart.
The total area bounded by the polygon, the extreme ordinates,
and the axis of x is practically

=sum of the given ordinates

=2i»(Hir

=1024.

. .)x(l)

The equation of the normal curve is

where (T^=npq

and

llXiXf:

•75,

Yo=N/\/27r • a=1024/V(5-57r).
Hence, taking logs, we have

X'

log ?/=log Yo-—, logioe
Zct"

=2-3915437-a:2(0-0789626).

It is easy from this equation to calculate the normal curve ordinates
corresponding to x=0, 1, 2, 3, 4, 5, and the results, compared with
the polygon ordinates, are as follows : — -

THE NORMAL CURVE OF ERROR

233

X

Ordinate of Polygon.

Normal Curve Ordinate.

0

252

246-3

±1

210

205-4

±2

120

119-0

±3

45

48-0

±4

10

13-4

±5

1

2-6

Now although the circumstances in which the series

{\Y^n{\Y-\\)-

.M^-l)a,n-

1.2

{\Y-\W^

may be taken to represent the frequency distribution resulting
from a particular kind of experiment were so stringently defined,
there is no reason why the normal curve itself to which the theory
led should be subjected to precisely the same limitations. After
all, the real and only justification for choosing one curve rather
than another to fit any given observations is that it does succeed
in fitting them better. But when the further question is asked
why the normal curve should succeed in describing some results
so well, we must not be tempted by analogy to rush to the con-
clusion that the causes at work are necessarily independent, and
equal, and so on. In short, the theoretical justification and the
empirical use of the normal curve are two quite different matters.

Experience shoAvs that the normal curve suffices to fit certain
types of distribution, besides those which arise in tossing coins and
in similar experiments, with remarkable accuracy ; among these
may be noted : —

1. Certain biological statistics ; for instance, the proportions of
male to female births taken over a series of years for a large com-
munity such as the population of a countr}^ ; also the propor-
tions of different types of plants and animals resulting from cross-
fertilization.

2. Certain anthropometrical, jJdrticularli/ craniometrical and allied
statistics, such as the height, weight, lengths of various bones, skull
measurements, etc., of a large group of persons, and the agreement
is the closer if the group be reasonably homogeneous, i.e. composed
of individuals of the same nationalitj' and sex between the same
narrow age limits, etc. ; also measurements of a similar character
in animals and plants.

3. Errors of observation in experimental work ; for example,

234 STATISTICS

several measurements of the same quantity — length, weight, speed,
temperature, or whatever it be — will contain errors of this kind
which are equally liable to be above or below the true value.

4. The marks of shots upon a given target, assuming that the
shots are equally liable to err in any given direction. This is an
interesting case of the normal law in two dimensions, for the north
and south line and the east and west line through the centre of
the target may both be regarded as axes of normal curves of error.*

5. Certain sociological statistics of a comparatively stationary char-
acter ; for example, rates of birth, marriage, or death at neighbour-
ing times or like places ; also the wages (and possibly the output
if it could be satisfactorily measured) of large numbers of workers
engaged in the same occupation under the same general conditions.

6. Any statistics or quantities that are individually compounded of
a large number of elements, mostly independent of one another, which
themselves vary between limits not very widely divergent, and none
of which exert a preponderating influence upon their resultant
statistic. The latter may be simply the sum of its elements, or,
more generallj^ it may be any function of the elements which, to
the first degree of approximation, can be expressed in linear form.

Now it would be a difficult matter in most of these cases to satisfy
ourselves as to the fulfilment or non-fulfilment of conditions like
those on which the binomial distribution rests. It is not easy
indeed to visualize them perfectly, except in artificial experiments
where they are largely under control. If anything, the chances
seem almost hopelesslj^ against their fulfilment in ordinary life,
so closely must we hedge round our sample to keep out unequal
influences. For example, to use a frequently quoted illustration,
if p measures the chance of death for an individual, the death rate
varies, as we know, considerably from place to place according to
the age and sex constitution of the population ; it is influenced by
differences in class, and occupation, and manner of life ; it is
altered from time to time, violently by the ravages of war or disease,
more gradually by improvement in general sanitation, housing
conditions, etc. We should only expect to get the binomial distri-
bution (and consequently the normal law if it depended upon the

[* Sir John Herschel published in the Edinburgh Review (1S50) an a priori
proof of the normal law from a consideration of this problem. Taking <p(x^) as
the expression of the law for one dimension and ip{x'' + 1/"^) for two dimensions,
the independence of errors in perpendicular directions leads to the functional
equation '^(x- + y'^) = ^(x') x <^(jr^), the solution of which is of the form

^/-g-?) _ g - A2j-2^ It should be added that the assumptions underlying the proof

s'tt
are not entirelj' above criticism.]

THE NORMAL CURVE OF ERROR 235

same postulates) exactly verified if we were dealing with the same
stationary population existing under the same stable conditions
over a long period of time ; moreover, since p is to be identical for
each individual event in the ideal case, it would be further necessary
that every family and every individual in our population should
also remain in the same stationary and stable state. This is mani-
festly impossible, especially after the industrial revolution which
the advent of machine power created.

These considerations suggest the interesting question whether the
various types of statistics we have enumerated, as being approxi-
mately subject to the normal law, could not, if we knew more
about them, really all be included under heading number (6), repre-
senting a further development from the binomial theory and an
enlargement of the field in which it holds good.

In an earlier chapter, when we were discussing the connection
between marriage rate and prices, we showed how it was possible
by a method of averaging to differentiate between long-time and
short- time effects. The more transient fluctuations, only super-
ficial in character, were removed and the real nature of any per-
manent change in the figures was revealed. In much the same
way, when we have a group of statistics which do not perhaps fit
a normal curve of error at all closely, it may be possible by random
averaging to get rid of some of the fluctuations which cause the
badness of fit and to obtain a new group of statistics which more
nearly obey the normal law. Averaging, that is to say, tends
to smooth away the rough outstanding abnormalities ; and we shall
presently show that if two variables, X^, Xg, which are independent,
obey the normal law, any linear function of the variables
(WiXi+w;2X2), obej's the same law. This may throw some light
on Class (6) where each statistic represents a compound, that is,
in a broad sense, a kind of an average of a large number of elements
which partially neutralize one another's influence, or rub the corners
off one another, so to speak, since no single element is, by hypothesis,
to exert an overwhelming influence upon the compound itself.

But although the normal curve does serve to describe a consider-
able number of frequency distributions within reasonable limits,
there are many more cases in which it fails : for example, the
greater part of those bearing on economic matters ; also statistics
relating to the incidence of disease and degree of fertility are, as
a rule, very markedly skew. Hence arose the necessity for an
extension from the symmetrical normal to some kind of skew
variation curves to fit such distributions.

236

STATISTICS

The normal curve, however, has an importance of its own to
which we must now draw special attention. It is the foundation
of the theory of errors and provides us with an invaluable method
of estimating the importance of one error in comparison with
another, or of determining the probability that an error shall lie
between stated limits. Upon it we depend for several most
important approximations which are in constant use.

The term ' error ' is used here in the sense that if we take the
mean ot a number of observations, the deviation of any one of
them from the mean may be termed its error. When such devia-
tions can be satisfactorily fitted, that is, within the limits of random
sampling, by means of a normal curve, they are said to be subject

to the normal law of error.
This law is expressed, as we
have seen, by the equation

where y . hx measures the fre-
quency with which an observed
organ or character deviates from
tlie mean by an amount lying between x and {x-\-hx) in a large
population, i.e. y . hx registers the frequency of an error of size x
to {x^hx), and N and a are constants dependent upon the particular
application of the law.

The probability curve or nortnal curve of error. As a guide to the
drawing of the above curve it may be worth while plotting

y=e-^.
This is readily done by writing the equation in the form

—x^=\og^y.
Giving now to y the values U, 0-1, 0-2, etc., we can find values of
logg y as shown in Table (48), and, by means of a square root table,
X is then determined.

Table (48). Corresponding Values of x and y to plot ?/=e~*\

y

logey

X

±00

y

loge^

-0-5108

X

±0-71

0

— 00

0-6

01

-2-3026

±1-52

0-7

-0-3567

±0-60

0-2

-1-6095

±1-27 '

0-8

-0-2232

±0-47

0-3

-1-2040

±110

0-9

-0-1054

±0-32

0-4

-0-9163

±0-96

1-0

0

0

0-5

-0-6932

±0-83

••

••

THE NORMAL CURVE OF ERROR

237

This enables us to plot the graph as shown in fig. (40). Since
logg 1=0, and the logarithm of any number greater than 1 is
positive and thus cannot be equal to —X', it follows that y cannot be
greater than 1 . Moreover y cannot be less than 0, for the logarithm
of a negative quantity is meaningless, but, as y approaches 0,
X approaches OD.

Also the curve is symmetrical about OY because for any possible
value of y there are two values of x, equal and opposite.

Returning now to the curve

y-

N

- 6-^^/2-=',

V27T.a

it must be of the same general shape as y=e~^ because the two
only differ in their constants. It is clearly symmetrical, for

»>

Y

1 1

_-_.__ _ _J. lO - - .

4.22« +

__, = _TT_=.,__J

.__! tiC Q. 75 >»

:::::::--::-::::::-:::::::: 5?! ::::::::: T": :::-: "i: :: ::

.. __. _ ^^ - .-± -5*..-

_ -'_ ... 41- . . 5,.

:::::::::: ::::::::_i't: ::::::::;::: :::±t:::::::::::::::::'i

:::::::::; :::;";!:^:i;::::::::::::::::$ 25;:::;::::::::;:::;:

Ji.;:j:::::::::::;

:± : :::::?. _ _ XX- - x

±: ..,,!f- _ _ .± . _ ± x

^s.-^- — ^ xii

H||||||nWtl#4JX

•200 -1-75 -150 -1-25 -100 -0-75-0-50 -0-25 O 0-25 0-50 0-75 100 125 1-50 1-75 2 00

Fio. (40). The graph of y = e-='\

instance, about the axis of y, because, in this case also, to any value
of y there are two values of x equal and opposite. Moreover it
tails off to the right and left from OY, the axis of x being an
asymptote, for as x tends to ±00; V tends to zero as before.

When

x^O, y=NlV27T.a,

giving the point B, fig. (41), where the curve cuts the axis of y,
Tliis is evidently the highest point on the curve, for

dy
dx

and this vanishes when .r=0.

d-y

Nx

,-a-2/2<r3

V2.

\TT .a"

Again,

^^g-*2/2<rI

■1 + '

dx' ^/l-na^ \ a-

which vanishes when x = ^(j, and at these two points, H. H', we

238

STATISTICS

therefore have ' points of inflexion ' where the bend of the curve
changes its direction.

The axis of y about which there is symmetry evidently locates
the mean error, in this case zero ; in fact the mean and mode
coincide, so that the mean or zero error is also the one which most
frequently occurs, and any two other errors which are equal in
magnitude but above and below the mean respectively occur with
equal frequency : i.e. the frequency of positive errors is balanced
by the equal frequency of negative errors on the other side of the
mean, making the median error likewise zero.

Again, the area I ydx measures the frequency of errors lying

J + Xl

between x^ and 0:2 above the mean ; I ydx registers the frequency

J-x
Y

Fio. (41).

of errors between 0 and .r, or of deviations up to this magnitude,
on either side of the mean ; and, in particular, for all errors

the total frequency = I ydx

J -CJO

= ^ f" e-''!''''dx

-\/2-Tr rrJ-

'V27r . CT-'-co

N

{V27T . ff) (as on p. 206)

V27r-CT

This enables us, by means of the fundamental definition, at once
to write down the probability of errors between any stated limits
and explains the origin of the name, the probability curve, which

THE NORMAL CURVE OF ERROR

239

is sometimes given to the equation. Thus we hav^e the probability
of an error between -{-x^ and -\-X2

frequency of errors between the given limits

frequency of all errors

Jxi

.+00

= ydx n ydi

-g-x'W^^jj .

. (1)

VStT . CT-'xi

Incidentally, the probability of an error between x and

N

Sx

-\/27r.(

e-x2>2

. (2)

Fio. (42).

Geometrically, the area represented by the shaded portion of

fig. (42) measures the frequency of errors between -j-Xj and -{-x^,

while the complete area between the curve and axis X'OX measures

the total frequency, so that the probability of an error between

-f Xi and -\-X2 is measured by the proportion which the area of the

shaded portion bears to the whole area.

dx
If in the above expression (1) we put x/a—^, so that — =ct,

di

it becomes

1 /-fa

(3)

which is known as the 'probability integral, ^^ and ^0 being the

240

STATISTICS

values of | which correspond to the values x^ and x^ of x. But
this integral measures the area of the shaded portion of the curve

1

y-

■hP

V27T

(4)

shown in fig. (43), which is really the normal curve over again, but
drawn on a different scale, namely, with the ordinates reduced in
the ratio N : a and with the standard deviation a taken as the
unitof measurement for a:, for I =1,2, 3 . . . when.r=CT, 2a, 3a, . . .
This has the effect of making the total area unity and the area
given by

a-,

d|

(3) bis

V27r4

now directly measures the probability of an error between af i and af g-
y Tables have been prepared

(see pp. 284, 285) which enable
us to write down the value of
this integral for different values
of fi and ^2 between certain
limits (see Appendix, Note 10).
Let us take an example to
show how the curve may be
used, and we choose one leading
to a binomial distribution, so
giving an expression for the
probability by first principles,

in order to compare the two methods.

Example. — Suppose we toss simultaneously 100 coins, and sup-
pose the chance of success, say ' heads,' is the same for each coin
and equal to 1/2. In that case, according to the binomial theory.

the probability of 100 heads ={l/2)^^^,

„ 99 heads and 1 tail =iooCi(l/2)99(l/2),

„ 98 heads and 2 tails =10002(1/2)98(1/2)2, andso on.

The most probable number of heads =np = (100)(l/2) =50. This
does not mean, as explained before, that if we perform the
experiment once we are sure on that one occasion to get exactly
50 heads and 50 tails, but that if we go on repeating the experiment
we shall in the long run get 50 heads and 50 tails turning up more
often than any other combination.

Let it be required to find the probability of getting at least 55

THE NORMAL CURVE OF ERROR 241

heads, that is, we want the probabihty of gettmg 55 heads or
more, and this is given by

a sum not very readily calculated if we have to go at it in a straight-
forward manner.

Now let us turn to the curve of error method. The standard
deviation for the distribution is given by

Since the mean number of heads to be expected if the experiment
is repeated a considerable number of times =50, we want to find
the probability of an error equal to or greater than 5, i.e. an error
lying between a and 4-00> because ct=5.

But the probability of an error between a^^ and cr^g

V27T.'h

Hence the required probability
1 -^

\/277.'l

=0-15866, by the probability integral tables.

In other m ords, if we repeated the experiment 100 times, we might
expect 55 or more heads about 16 times.

We can now show that if Xj, Xg are two uncorrelated variables
obeying the normal law, then {w^-^-\-w.^<^) will obey the same law.

Suppose Xy, Xo are observed deviations from the mean values
Xj, X2 in one particular record, ctj, o-g being the respective S.D.'s.

Let X=?rjXj+w'2X2' ^^<i ^6^ •^' ^6 t^^ de\4ation in X corre-
sponding to deviations x'^, x^ in the given variables.

Thus X-|-a:=i<'i(Xi+.ri)-l-W2(X2+'^'2)

Therefore, x—w^x^^w^x.^,.

But the same error r may be obtained by giving x^, t, many different
values provided their weighted sum is unaltered. Let us first
keep x-j constant, so that the corresponding value of x^ required

Q

242

STATISTICS

to produce an error lying between x and {x-\-hx), where hx is small,
must be such that

X < 10 yX^ + ^2^*2 < -^ + S-^j

i.e. x—w^x■^^<iWoX2<.x—WyX^-\-hx,

i.e. X2 lies between (.r— w^.rj)/^, and {x—w^x^-\-hx)jW2i and the
probability for this

=?f . __L_e-(--'">^^)^/2-^2'-'.^ by (2).

Now this is in a form which only involves hx, x, and x^, and we
get the total probability for an error \y\ng between x and {x-^hx)
by givmg all possible values to the error x■^^.

But the j)robability for x^ itself to lie between x^ and {x^-\-hx^

_ 1 r^i+^-'-i

g-x-/2<r2j^^

Sxj

V27

-e-^'^/--S by (2),

f _ ^ e-(a;-«-iJ-iy^/2g-v.'^2

'2 a2\/27r J

and the probability for this to concur with a suitable x^ to produce
an error in the weighted sum lying between x and (a:+Sx*), on the
assumption that X^ and Xg are independent, is therefore

hx

W,^TTa-^2

Hence the total probability for an error lying between x and (x+Sa;)
is obtained by integrating this result, that is, summing all possible
probabilities, between .Ti = — CD and Xi =+oo. This gives

"•*' / g \2(r-i 2o-22W-''2' •1<t\\k\ 1<j\w'^^^

lO'2.'-o3

hx

1W\X

— ^1-

rtJg • 27rcriCr2- - =0
where a^^W'ya'^■^;Y'^'•p''^

^j, .4-00 _,/ crJi _ U'i^i ^\ = ^^2/ cr^iwiii 1_ \

1^2 • 27rCTjCr2.
So;

rfa*!

a2(ofli£Vi^)r .+CO

^2 • 27rcriCT^

g 1<jSw\<t-^

I — ry-i

\/2 . 0'iO'2W^2

THE NORMAL CURVE OF ERROR 243

1 / CT.X-1 It'iCTj

where f=—-=\ — x

•V2\CT|0-2^2 0"2^20"

8^ ,-'-..^&^^V2.c7,a3ii;3

^2 • 27rcriO-2

\/27r.CT
which proves that the error x obeys the normal laAv with

S.D.=V(wVi+wV\) .... (5)

The above principle is readily extended, for if

X^, Xj, . . . X„ being independent variables obeying the normal
law, then X also obeys the normal law and its

S.D.= V(wV\+wV2+ . . . +wVJ . . (6)

In discussing the results of random sampling we worked upon
the principle that, given a number of sample observations of any
statistical constant, a mean or a percentage or a coefficient of
regression or anything else, an error or deviation as large as cr,
the standard deviation, from the true value for the whole population
might quite likely occur, but that an error exceeding 3cr would be
unlikely, and we explained that, as a result of convention, the
probable error, equal to §ct roughly, was largely used in place of a
by many writers. We have now to examine the basis of this
principle, and the first point to notice is that it only strictly applies
to a normal distribution.

To find the probability of an error lying between —a atid -\-a in a
normal distribution.

1 /"+' . .-,
The required probability = — -= — I e~'^^'^'^'dx

V27r . a- -<r

1 r+i
=—7= I e~^'H^ (where x=a^)
V27r-'-i

=0-6827, by means of the tables.

This then is the probability that the error in a given sample shall
not exceed the S.D., a. The probability that the error shall exceed

244

STATISTICS

a iB accordingly (1—0-68) =0-32. It therefore appears that the
odds against an error exceeding this amount are 68 to 32, or about
2 to 1.

The probability of an error between —2a and -\-2a

1 /■ + 2

V2J-2

=0-9545,

and the probabihty of an error outside these limits =0*0455.

Hence the odds against an error exceeding 2a are about 21 to L

The probability of an error between —da and +3a
1 r+3

-P/2,

di

V27TJ-3

=0-9973.
Hence the odds against an error exceeding 3a are about 370 to 1.

1

1

Zj

1

1

'

B

"T

--^

[

<'

"

1

/'

\

/

'S

n'

^^

1

V

__^

\

R

y

s-

J

kC:i

\

/

S

/

--

s

/

II

■ ^

V

'

Hj

*"s

■p»

s

F

CO

N

-3i

--

--

n

--

M-

-0'6-7

'it

"cr-

J-

-

_0-

4W

■=-

i^

^-

i^.—

O ^^^R N, <^ N2 <^ N3 X
Fig. (44).

That these results are reasonable can be seen by an examination
of the curve of error

N , ,

V 277 . a
the graph of which is drawn, fig. (44), in the particular case when
cr=5, N=.100. The maximum ordinate is thus=20/\/27r=7-98,
and the curve becomes

7/=7-98e-*'/5o.

When .r= a= 5, ?/=(7-98)(0-606)=4-84, P^Ni in the figure.
„ jc=2a=10, y=(7-98)(0-135)=l-08, PgNg „
,, .r=3t,=i5, iy=(7-98)(0011)=0-09, P3N3 „

THE NORMAL CURVE OF ERROR 245

There is a point of inflexion where the curve changes its
direction at P^, also at the companion point F\ on the other side
of OB.

The areas ONiPiB, ONgPaB, ON3P3B, P3N3X represent respec-
tively the frequencies of errors 0 to cr, 0 to 2a, 0 to 3cr, 3a and over
(considering only errors on the positive side, that is, deviations
above the mean), and the figure shows how very improbable is a
deviation from the mean exceeding 3a, for the area between the
curve and axis beyond this limit is negligible. Put in another
way, a range of 6a should include practically all the observations
in the sample.

The probable error has in the past received various names, such
as mean error, median error, quartile deviation, and although some
of these may seem more applicable and less confusing than the
name to which it has settled down, there is perhaps not sufficient
excuse for unsettling it again, even had we the power to do so,
by attempting a return to one of these old names.

If its magnitude be r it is defined to be such that the chance
of an error falling within the limits — r and +^ is exactly equal to
the chance of an error falling outside these limits, in fact it is an
even chance whether a particular error falls within these limits
or not.

Since area measures frequency it follows that the ordinates
drawn through the probable errors divide both halves of the normal
curve (above and below the niean) into two equal parts ; the one
above the mean, QR, is shown in fig. (44), and consequently the
area OBQR=the area QRX, in that figure. These ordinates there-
fore coincide with the quartiles, and the probable error is precis(ily
the same measure as the quartile deviation.

The magnitude of the error is readily calculated from the proba-
bility integral table, for, by definition, we have

1 r+r!<T
= , — I e~^~'H^ (where x=a^).

Hence i=^= e-^-'H^,

V27T.'0

and the probability integral table at once gives
r=0- 6745a = approximately |a.

246 STATISTICS

Thus we have the frequently quoted rule that the

quartile deviation =?(standard deviation), . • (7)

or probable error =0*6745 (S.D.)

The probability of an error lying between — 3r and +3r
1 ;+3'-

2 ,-3(0 -6:45)

= —^=1 e~^'''^d^ (where ?c=a^ as before)

V277-'0

=0-9570.

Thus the odds against a deviation exceeding three times the probable
error occurring in a single trial are about 22 to 1, or much the same
as the odds against a deviation exceeding twice the S.D.

There remains one other standard of measurement in connection
with errors which is at least deserving of mention, namely, what
we have previously called the mean deviation, which may be denoted
by ?;. It is simply the mean of all errors -without regard to sign ;
thus, since ySx measures the frequency of an error lying between
X and (x+Sx)

71 = 2 1 xydx / 2 I ydx

Jo /Jo

/■CO I rca

= \ xe-^^'^-^-dx e-'^'^-^'^dx

Jo I Jo

•CO / -oo

=(t| ^e-^''-dd \ e-^-'-di {where x=a^)

= -v/2ct / ^te- - ''dt I \ e - 'V< (where ^^ =2^2)

.'0 / .'o

=V2ff

2

e
_ ~2

^ctV2]^

=0-7979(7,

hence the rough rule that the

mean deviation = ^(standard deviation) . . • (8)

It must be borne in mind that all the above rules relating to
errors — using the term as synonymous with the deviations of single
or sample observations from the mean of a considerable number of
the same character — strictly apply, as we said before, to the normal

THE NORMAL CURVE OF ERROR

247

curve of error and are only approximately true for other distribu-
tions, the approximation being the closer the nearer they approach
to the normal form and the larger the number of observations
involved. They have been tested in some cases in earlier chapters
(see, for example, Chapter VII.), and the results obtained, even
with very skew distributions of comparatively small numbers of
observations, are at all events close enough to suggest the utility
of the rules in more favourable cases.

The effect of variability on errors. The probability of an error
lying between 0 and t

1 /-«

■\/2tt . Or.'o

Put x^^x'jm, and this becomes

V27T . CT.'i m

1 /•("")

.-x2/2(m<7)2

V2tt . {ma)

dx.

^

1 [

'-- - - Ti

1 1

1^ _i^tj

/r • t . ^

^ \ ' t ' ! \

f : • ■ : I ;\

J ! ' ' ■ ■ ' '■ \ h-.

.jL _j_ . „ >...^- . ... .

;z 1 J B, _ ^c:

7 ^ ;"] , 1 " ' \^ ....

jtoj-^f^ i ; ; I 4.._j — -■'■^Vl 1 f--

- - " ^-?

;<?-5^:::^:^:^.^-_ i : -^:.^<^;^:::::::::::::::::::

:;^'" i

1 X ■ : * ii-.

-s--' ''K

iiiii---^^ —

.^ H !---|rT^i-j/t-r^--— : -r^-r i rri--'4ii..--..::--:

Fig. (45).

N2
..2t---

Thus, if the variability be increased m-fold the range of error (of
equal probability) is increased m-fold, so that if we have two sets
of N observations, with the variability of one set double that of
the other, the range of error also in the one set is double that which
is equally likely to occur in the other. This is brought out fairly
clearly in fig. (45), which is the result of plotting the curve

y

N
V27rcr

248 STATISTICS

in the two cases. The variability a of curve (1) is double that
of curve (2) ; if then we measure along OX in the figure

ONi-20N2=2<,

the area BjONiPi will be equal to the area B2ON2P2, showing that
the probability for an error between 0 and 2t in the one case is equal
to the probability for an error between 0 and t in the other case.

[James Bernoulli (1654-1705), the eldest of three remarkable brothers,
showed how the binomial theorem could be used to estimate the probability
that the ratio of the number of successes to the number of failures under
defined conditions should lie between set limits, where success means that a
certain event happens and failure means that it fails to happen.

It was C4auss who first actually published a proof (1809) of the equation of the
normal curve, although Laplace had suggested as early as 1783 the utility
of a probability integral table, je-'-dt. Gauss's proof depended upon certain
axioms which cannot be established and are not necessarily true, one of which
was that ' errors above and below the mean are equally probable.' Laplace
and Poisson improved upon Gauss and succeeded without assuming this
axiom, but with the aid of theorems due to Euler and Stirling, in developing
the continuous probability integral from the discontinuous binomial series.

Further extensions of the normal curve applicable to skew distributions
have been worked out by other writers, such as Galton and McAlister, Fechner,
Lipps, Werner, Charher, Kapteyn, and finally by Edgeworth, who has contri-
buted materially to the development of the idea of ' the Law of Great
Numbers.' Karl Pearson approaching the subject of skew variation from
the same point but by an original route, has discovered a complete system of
curves suitable for fitting almost all kinds of distributions in homogeneous
material, especially such as are met with in the biological world.

(See Todhunter, History of Probability.

Edgeworth, Law of Error in the Encyclopaedia Britannica (10th edition).
Pearson, Das Fehlergesetz und seine V eraUgemeinerungen durch Fechner
und Pearson: A Rejoinder; Biomelrika, vol. iv., pp. 169-212).]

CHAPTER XIX

FREQUENCY SURFACE FOR TWO CORRELATED VARIABLES

It may serve at this stage to widen the outlook upon the subject
of correlation for those who are able to follow it up on mathe-
matical lines if we briefly consider the algebraical expression for
the combined distribution of two variables.

Let the variables be X^, Xg. They may be absolutely independent
or they may be related in some way, but in either case we shall
assume it possible to set up a one-to-one correspondence between
them : thus, X^ might represent the marriage rate and Xj the
index number for wholesale prices, and we might always pair
together the X^ and the Xg which refer to the same year, as in the
correlation example in a previous chapter ; moreover this pairing
might still be effected even if there were really no other connection
at all between X^ and Xj.

If then x^, x^ typify the deviations of Xj, Xg from their respective
means (the means in the above case being derived by averaging
the figures for a number of years), it is possible to write down an
expression of the form

y=F{Xi, X.,)

for determining the probability of deviations between a^j and
{x^-\-Sx^), x., and {x^-^rSx.^), occurring simultaneously (in the same
year, in the above case) ; or, to put the same thing in another way,
yBx^Sx.^ would represent the proportional frequency with which
such deviations might be expected to occur together in a large
number of observations.

The frequency curve y=f{x), where yhx denotes the frequency
with which a variable with deviation lying between x and (a:+Sx")
from its mean value is observed in a given distribution, was repre-
sented by plotting corresponding pairs of values of x and y as
points in a plane. In the expression y=¥{x^, x.^), however, we have
three variables to consider, x^ and x^, and j/ which measures the
frequency of the simultaneous appearance of x^ and x.^. Such a
trio may geometrically be represented by a point P {x^, x^, y) in

250

STATISTICS

Tig. (46),

space of three dimensions, for {x^, X2) can first be located as a point
in a fixed plane and a height y may then be measured above this
plane as in fig. (46). Clearly as .r^ and x.^ vsLvy, y also varies, and
consequently the point P moves about in space, but it moves always
in obedience to the relation

y=F{x^, Xg).

This relation is called the equation of the surface along which
P travels, showing that it holds good for
the co-ordinates (.r^, x.^, y) of any position
which the point can take up on that surface.
It is convenient, however, to use the notation
* z=F{x, y)

in preference to y=^{x-^, x.^ for the 'fre-
quency surface,' because OX, OY are nearly
always taken to represent the axes of refer-
ence in space of two dimensions {i.e. in a plane), and by a natural
extension OX, OY, OZ are taken to represent the axes of reference
in space of three dimensions, fig. (47).

We proceed to discuss the frequency surface for two variables,
and we shall start with the comparatively simple case when the
variables are completely independent.

Frequency surface showing distribution of
two completely independent variables each
subject to the normal law.

Let X, Y be the variables, and let x, y de-
note deviations from their means X, Y, the
point (X, Y) being taken as origin of co-ordi-
nates and the usual notation being adopted.

Thusthe probability of a deviation betweena: and (.t+ 8a;) occurring

hx

Fio. (47

, X3
J — 5
<Txr-

V2i

and the probabiHty of a deviation between y and {y+hy) occurring

Therefore the probability of such deviations occurring together
since the variables are supposed completelv independent

277'CT,CT„

- 3/-/2<^y

FREQUENCY SURFACE 251

Hence the frequency with which such pairs of deviations are
observed together if n be the total number of observations

Denoting this by zSxSy, we get for the required frequency surface,

-^(4+4)

z=nl2Tr(T^y . e "'' <'y'' , . . (l)

If we give y some particular value, t/j, we find from the above
equation that the law of frequency for the corresponding x is

-i(£?+?di)

2Tr(7j.Gy

. e

277(7 a-CT^

,-x2/2<r^

-^ c ,

V 277 . (7^

where n^ has been written in place of

\V27T.ay

But this is evidently a normal curve in the plane X^OZj, having
the same mean, X, and the same S.D., a^, whatever be the value

of 2/1-

Hence all arrays of X are similar, having the same mean and the
same standard deviation, and this, by symmetry, also applies to
all arrays of y.

Now put z equal to some constant, k, in equation (1), so that

k-

2TTGy.ay

n

Since the left-hand side of this equation is constant for different
values of {x, y), it foUows that the right-hand side is also constant,
and hence

^+4=«. • • • (2)

where c is a constant.

We conclude that the values of x and y which can occur together
with a given frequency, k, are such that the point {x, y) always lies

/

252

STATISTICS

somewhere on the ellipse (2) in the plane s=A% fig. (48) ; e.g. values
in the neighbourhood of .t^ and y^ occur with the same frequency as
values in the neighbourhood of Xg and 0, because in the figure the
points (xj, y^, k) and {x^, 0, A;) both lie on the ellipse defined by

7 ^" , V"

z=k, — +J^=c.

The different ellipses which can be obtained by varjdng the
frequency, and consequently varying c, are clearly concentric,
similar, and similarly situated if they are orthogonally projected
on to the plane s— 0, for the effect of such projection is that any

Fio. (48).

point (re, y, z) drops Aowa. on to the point {x, y, 0) which stands
immediately below it in the plane XOY.

The general shape of the surface can be gathered from fig. (48)
where the ellipse in the plane z=k, and the normal curves in the
planes a;=0, y—0, and y—y^ have been dra%vn.

It will also be noted that if the scales of x and y are altered by

X 11

writing — =a;' and —
Or a,,

y', so that unit change in each may be the

same, the ellipse (2) becomes a circle
x'-'^y''=c.

This change of scales is equivalent geometricallj'^ to projecting
orthogonall}' the ellipse into a circle ; of course the planes of pro-
jection are not the same as in the previous orthogonal projection
mentioned.

FREQUENCY SURFACE

253

Frequency surface for two correlated variables. Let the variables
be X and Y, and let us work as before with their deviations x and y,
whichis equivalent to taking the mean point (X, Y)of all the observa-
tions as origin.

Now the line of regression giving the best y, or the y of greatest
frequency, corresponding to any x is

y-

-r^'x,

with the usual notation, r being the coefficient of correlation
between X and Y.

Hence the error made in estimating anj- y from this equation
instead of taking the y given by observation is

V—y (observed) —y (estimated)
=y-r^x. [See fig. (49).]

Thus, corresponding to every pair of observations {x, y) there is
an 77, and the same rj Avill be repeated Y
just as often as the same pair of
observations {x, y) is repeated.

Therefore the frequency distribu-
tion of {x, Tj) must exactly correspond
to that of {x, y).

Further, the correlation of the
variables x and 7; is zero, for posi-
tive and negative errors r) are equally likely to occur for different
values of x\ in fact, this coefficient of correlation is 2J{xT})/na3XT^^, and

Fio. (49^

2{xi])=I!

x[ y-r

O-x /J

=Z{xy)-r^-^.S{x'')

■=np'

P

no.

—np—np

=0.

Assuming then that the variables x and r) are quite independent,
the probability of them occurring together is readily written doAvn,
for it is simply the product of their separate probabilities.

254 STATISTICS

But the probability of a deviation between x and {x-\-8x) occur-
ring, if we consider this variable alone, is

-e

Sx

V27

and the probability of a deviation between rj and {r]-\-hi]) occurring,
if we consider this variable alone, is

6v ^ 2.,^^

A/27ra,
Hence the probability of a combined occurrence of such deviations

V27Ta^ /\V27TG^

27rCTa.a.
0x8 y]

-e

But na^'=2:iy-r^x

=w<T„2(l-r2).
Similarly, na^^=na^{\—r^),

where ^ is the error made in estimating x from x^=r—y\

Thus

Oy _ I CT„_ 1 (Ta;_ 1

FREQUENCY SURFACE 255

and il,+ I^)=Lil+r'-<

=__ 1+r-. )

1

or,^(l-r2)

1

Hence the probability of the combined occurrence of deviations
X to {x-\-Bx), Tj to {t]-\-87))

2770-a. . (TyVl — r^

thus, if we denote by zBxSy the frequency of the combined occur-
rence of deviations x to {x-^hx), y to iy-{-8y), when w is the total
number of observations, we have *

27r-\/l— r'^ • o'xO'i/

When the variables X and Y are completely independent, so that
r is zero, this reduces, as it should, to our previous result

n

e

-■iV^"-|- ^' _2r '^^^ '\ ^

In the .surface z=fi . e '^"^^^ <^i'* <ri<ry/i-r2 ^ , . (3)

where />t = == , if we give y some particular value t/j,

27TVl — r- . a^Uy

we find that the law of frequency for the corresponding x is

__JL_*^(l-r2)+('£-,M*}
=U,.e 2(l-r2)|o-/ '^V.T. <r,/ i

=/ti . e

=ju. . e

3/% 1 / X _yyi\2

"Wg 2(1 -rzA^ ^/ ^ , ^ (4)

[* For an outline of Karl Pearson's method of reaching the Law of Frequency
for two correlated variables, and certain deductions from it, see Appendix,
Note 11.]

256

But just as

STATISTICS

(r-a)2

1 -i

y = = — ^

represents exactly the same normal curve as

shifted through a distance a along
the axis of x, fig. (50), so we con-
clude that the curve (4) in x and
z, in the plane y=yi, is exactly the
same as the normal curve

gV=l/2<r,;-

^e V.2(l-r2)

shifted through a distance ry^— along an axis parallel to OX. In fact
(4) represents a normal distribution for x, the mean, corresponding
to greatest frequency when 2= ^„^ ,^, being determined by the
intersection with the surface (3) of the planes

y=yv

.y

and the standard deviation being CTj,V1 — ^', which we note is
independent of y^, fig. (51). To put the same thing in another
way, the array of x's corresponding to a particular value y^ of y

have a mean deviating from X by r— . y^, and a standard deviation

a^Vl — r-.

In particular, when y=0, 2=yu.e '''^'^'^-'^-\ a normal distribution
for X, the mean, corresponding to greatest frequency with z^fi,
being determined by the intersection with the surface (3) of the

X J/

as before.

Similarly, when x=^Xy, we get as in (4) a normal distribution for y.

planes 2/=0, —=i—, and the standard deviation being a^vl—r^

get as in (4) a norn

2<rV-' 2fl-r2)\o-„ a J

^

the mean, corresponding to greatest frequency when 2;=--^-^^, being
determined by the intersection with the surface (3) of the planes

y X
x=Xi, — =r— ,

FREQUENCY SURFACE

257

and the standard deviation being ayVl—r-, which is independent
of Xy In other words, the array of y's corresponding to a particular

value Xj^ of x have a mean deviating from Y by r—x^, and a standard
deviation ct^VI — /".

In particular, when x—0, z=^^e <'^'<^-'''\ a normal distribution
for y, the mean, corresponding to greatest frequency with z=/j,,
being determined by the intersection with the surface (3) of the

planes .r=0, ——r—, and the standard deviation being GyVl — r''.

By putting 2=some constant, k, and arguing just as we did in the

2

Fio. (51).

case of two independent variables, we find that all values of x and y
which occur together with the same frequency define points {x, y)
which lie on the ellipse

z—k,

+^— 2/--

-c.

The different ellipses which can be obtained by varying the fre-
quency, and consequently varjdng c, are concentric, similar, and
similarly situated, if they are orthogonally projected on to the
plane 2=0, The planes giving the means of the x's, or the most
frequent x's, corresponding to particular values of y, and the means
of the y's, or the most frequent y's, corresponding to particular
values of x, meet z=0 in the lines of regression

l=,l.

y X

Q^ ffj.

CTj. O,

25S STATISTICS

X V

If we alter the scales of x and y by writing — =x' and —y'

so that unit change in each shall be of the same magnitude, the
frequency surface takes the form

1

" o?r — 74^^ ■-'+!/'- - 2rj;';/')

2=/xe -(.' "^^

1

When y'—O, z=iie. -^^~'''^ , a normal distribution, the mean being
on the plane x'=ry', and the standard deviation being Vl—r'^.

Similarly for a;'=0. When y' ::=y\, z=fj,e e ^(i-^^) i ^ ^
normal distribution, the mean being on the plane x'=ry', and
the standard deviation being Vl — r- as before. Similarly for
x'=x\.

Again the ellipse which is the locus of the points {x'y') obtained
by putting 2=constant, k, corresponding to variables which occur
with the same frequency, is (in the plane z=k) now

x'-^y'-—2rx'y'=c,
and, projecting on to the plane z=0, the lines of regression are

x'=ry', y'=rx'.

These lines are the intersections with z=0 of the planes containing

the means of the x"s, or the most frequent x"s, corresponding to

particular y"s, and wee versa.

X v

Since, geometrically, the transformation — =x', —=3/', is equiva-

lent to an orthogonal projection, we may learn something about
the more general ellipse by considering properties of the simpler
projected curve which are not changed by projection.

Let us first, however, find the magnitude and direction of the
axes of

x"--]-y''-—2rx'y':=c.

By turning the axes through some angle 6 this equation is
reducible to the form

which is the ordinary form for an ellipse when its axes lie along
the axes of co-ordinates. But the equation in x' , y' is clearly
symmetrical about the lines y' ^=x' and y' =^—x', because y' and x'
or y' and —x' can be interchanged without the equation being
affected. Hence these lines must give the directions of the major
and minor axes.

FREQUENCY SURFACE

259

To turn the axes of co-ordinates through an angle of 45°, fig.
(52), we must write

x =x" cos 4:5°— y" .sin 45

o x"-y"

V2

x"-\-v"
y'^x" sin 45°+?/" cos 45°=' ^"-,

V2

Fig. (52).

The equation of the ellipse thus becomes

2r

{x"-y"f , {x"^y"f ^S^"-y"){x''+y'')

2 2 ^2^/2

x"- + y"-- r{x"- - y"- ) =c,
a;"2(l-r) + 2/"^(l + r)=c,

=1.

\-r 1+r

=c,

I.e.
i.e.

i.e.

Hence the semi-major axis is a= / , and the semi-minor axis

•^ 1 — r

c c

is 6= . / . We note that as ?• increases from 0 to 1. a increases

1 + r

from Vc to 00, while b decreases from Vc to ^ / - . Also, as r decreases

from 0 to —I, a decreases from Vc to , / - -, while b increases from

V 2

Vc to 00.

260

STATISTICS

The ellipses, x'~-\-y"-—2rx'y'—c, corresponding to different values
of r all pass through the points of intersection of
x''-\-y"^=c and x'y'=0.

But x"^-\-y"~==cis what the equation of the ellipse becomes when r,
the coefficient of correlation, vanishes. The connection between
these curves is shown in fig. (53), which represents their projection
on to the plane 2=0. A positive correlation between x and y
might be expected to increase the y corresponding to a particular
positive X, if the frequency be fixed beforehand, and that is the
effect which the figure also would suggest.

Fio. (53).

Now, in x'--\-y"^—2rx'y' =c,

the lines of regression are

y'=rx', y'=-x',
r

and the axes of the ellipse are

y'=x' , y'^—x.

Hence the lines of regression are equally inclined to the axes of the

ellipse as well as to the axes of co-ordinates, fig. (54).

Further, the pair of lines

y'=x\ y' =—x'

form a harmonic pencil with the pair

.r'=0, i/'=0,

and also with the pair

, , , 1 ,

?/ =zrx , y =-.e .

r

This is obvious from fig, (54).

FREQUENCY SURFACE

Now project back to the ellipse

xy

The algebraical transformation for this is merely

2r"^ = constant.

x'^-, y' = ±

261

Y

V

\ ,

^--

-/I,/

V

/

^\\

/^l

X'

Fio. (54).

Since the harmonic property is unaltered by projection we then
have the pair of lines

(Ty d^ Oy CTa

harmonic with the pair

.r=0, y=0,
and also with the pair

y X y \ X

Hence the two lines of regression corresponding to maximum
correlation {r = -\-\ and r = — 1) are harmonic with

(1) the axes of co-ordinates ;

(2) the lines of regression for any r.

Again it may be easily seen that the lines
y'=rx' and x'=0
are conjugate diameters of the ellipse

x'-^+y'-^-2rx'y'=^c, , . . (5)
for they may be written as one equation thus :

rx"^—x'y'=^0,

262 STATISTICS

and this represents a pair of lines harmonic mth the (imaginary)
asymptotes of (5), namely, with

x'^-\-y'i—2rxy'^(i.
[The criterion for ax--^2hxy-\rby-=0

to be harmonic with a'x'-\-2h'xy-\rb'y-=^0
is ab' -\-ha' =2hh' .']

But it is a well-known property of conies that any pair of lines
harmonic with the asymptotes are conjugate diameters of the
conic.

Similarly it may be shown that the lines

y' =-x' and rj' =^0
r

are conjugate diameters of the ellipse (5).

But, on projection, the conjugate property also is unaltered.

II X

Hence the lines — =r— , .r=0,

y 1 X
and the lines — = , y^O

are conjugate pairs of diameters of the ellipse

X- , y- ^^ xii

But for conjugate diameters the midpoints of all chords parallel
to either lie on the other.

Thus we come back again by another route to the familiar line of
regression theorems that, for a given r, all arrays parallel to x=0

V X

have their means on— =r— , and all arrays parallel to y=0 have

., . X y

their means on — =r— •
a» a,,

APPENDIX

1. Compound Interest Law. If the capital increases continuously,
instead of going up by jumps at the end of stated periods, the con-
nection between the original principal S^,, the rate per cent, per
annum r, and the amount Sj at the end of i years is given by

for the rate of increase is measured by

dS_ rS
dt~]M'

which leads at once to the above equation on integrating.
Other instances of the same law are : —

(1) .4 particle moving against a resistance j^roxiortional to its
velocity, t\=Vge~''\

where i\ is the velocitj^ at time t, v^ is the original velocity, and c is
some constant.

(2) The variation of the pressure of the atmosphere ivith height,

where pJ^ is the pressure at height h above a surface level, p^ is the
pressure at the surface, and c is some constant.

(3) The rate of cooling, d,=d„e-'',

where 6f is the excess of temperature at time t of the hot body
over that of surrounding bodies, 6^ is the excess when the measure-
ment begins, and c is some constant.

2. Weighted Mean. Let the observations be represented by the
different values, x^, .r.,, . . . x„, of the variable concerned, and let
the respective weights attached to these observations be /i,/2, ■ ■ - fn^
so that the average, by definition,

A+72+ •••+/„ '

264 STATISTICS

Now, suppose a different set of weights be chosen, namely.
/'i./'2' • • • f'lv giving a ncAv average

•^ij l'T^2j 2 1 • • • I '^nj n

/'l+/'2+ • • ■ +/'.

The difference between these two expressions
/1+/2+ • • • /'i+/'2+ • • •

{/1+/2+ • • •)(/'l+/'2+ • . •)

]{fJ'2i^\-'^2)-f2f\{Xi-X2)\-\-\fJ'3{-Xl-Xs)-fJ\{Xi-X^)\+ . . .

(A+/2+ • • •)(./'i+/'24- . . .)

^ (/1+/2+ • • ■){f\+r2+ ■ ■ •)

Hence this difference is very small and the averages are very
nearly equal if the weights /j, f^, f^ ■ . ■ are replaced by others
f\, f'2, f's ■ ■ • very nearly proportional to them, so that fjf'i,
fjf'o, Jzlf'z ■ ■ • ^^® ^^^ ^^^ from equality, and this is the more
pronounced if the observations x^, x.^, x^ . . . themselves are all
of the same order of magnitude and the sums of their weights,
UfandUf, are large so that the expressions of ty])e{Xi—X2) /{I!f){I!f')
are small.

3. Geometric and Harmonic Means. Given ?i numbers
a, b, c . . .
their geometric mean, g, is defined by the formula

g=':yiabc . . . ),
and their harmonic mean, h, is defined by

h a 0 c

We note that when

a=b=c=: . .

. =k, say,

then

g — lj{kkk . .

.) = l/{k-)=k.

and

h k k k

n
' ' ' ^k

so that

h=k.

APPENDIX 265

It is worthy of remark that if the geometric mean be adopted as
average in discussing the index numbers of prices it possesses an
interesting property which does not hold for any of the other means
in common use.

Suppose the prices of n standard commodities at three successive
dates be represented by (aj, b^, Cj . . . ), {a.^, b^, c, . . .). (c^3^ ^3> C3 . . .).
Then the index numbers of the separate commodity prices at tlie
third date, taking the prices at the first date as standard, are

100'^^ 100-^ lOO'i . . .
«! b, c,

Hence the geometric mean of these n index numbers together
V \ Oi h Ci

= 100gr3/j7„

where g^, g^ denote the geometric means of the u prices at the two
dates.

It follows that the ratio

index number of prices at 3rd date with prices at 1st date as standard
index number of prices at 2nd date Avith prices at 1st date as standard

_1Q06^3M
lOOgJg,

=93/92-
It is therefore quite independent of the particular date chosen as
standard.

4. The Mean of Combined Sets oi Observations. (I) Suppose one
variable x is expressed as the sum of a number of other variables,

thus x=a-\-b-\-c-\- . . .,

and suppose that we have n different values of the variables, giving
equations of the type

a;2=a2+^2+C2+ • • •

a^n=an+^« + C„ +

266 STATISTICS

Hence, by addition,

^-l + '^•2+ • • +-^'„ = K+ • • +«J+(6i+ . . +6n)+(Ci+ . . +cj+ . .
so that nx = nd-\-nS-\-nc-\- . . .

x—d-[-b-\-c-{- . . .,

where x, a, b . . . denote the means of the n values of the respec-
tive variables.

Thus the mean of a sum equals the sum of the means, and, if some
of the positive signs in (a+6+c+ . . .) are made negative, there
will evidently be a corresponding change of sign in (a+^+ • • •)•

Example. — Suppose 100 family budgets are collected and the
items in each are separated under five heads — rent, food, clothes,
coals and light, sundries. The expenditure, x, in each budget would
thus be expressed as the sum of five variables, a, b, c, d, e, and the
mean of the 100 dififerent x's ^A'ould equal the sum of the means of
the a's, the 6's, the c's, the rf's, and the e's.

(2) Sets of observations are made which differ in locality or time or
some other respect. To find the resultant mean.

Let I observations of the variable x refer, say, to one date,
,, m ,, „ „ „ ,, a second ,,

,, n ,, ,, ,, ,, ,, a third ,,

and so on, and let the means of these successive groups of observa-
tions be Xi, x^^, x'„, . . . , so that we may MTite

Xi=ZxJl, x^^ExJm, x^^SxJn, . . .
If then X be the resultant mean, we have

l-\-7n-{- . . . l-\-m-{- . . .

Exajnple. — If the school children in the different schools of a
county are weighed, I children in one school, m in another, n in
another, and so on, giving mean weights x^, x^^, x„ . . . , the
resultant mean weight for the children in all the schools combined
is then given by the above expression.

5. Me^n and Standard Deviation of a Distribution of Variables.

Let x^, x.^, x.^ . . . x^ denote the deviations of each value, or group
mid- value, of the observed organ or character when measured from
some fixed value, and let f^, f^, fz ■ • • fn denote the observed
frequencies of these respective deviations.

APPENDIX

267

The arithmetic mean of the variables is thus given by

^ = (/l-^-l+te+ • • • +/n^n)/(/l+/2+ • . • +/n),

referred to the fixed value as origin.

We may conveniently represent the deviations x^, X2, x^ . . • by
lengths measured from an arbitrary origin 0 along a straight line,
in which case the point 0 defines the position of the fixed value
from which the variables are measured.

Let P mark the position corresponding to a typical variable and

let G mark the position corre- _ ^ e ^

sponding to the mean, x. Thus i: — '■ ± i

OP=a*, 0G=^% and if we denote ^ -^" ' ^

the distance of P from G by ^, we have

x^x+^.
Hence

^■ = (/A+/2'f2+ • • • +/n^-n)/(/l+/2 +
Hfl{^^-^l)+M^ + U+ ■ ■ • +fn{£-
= [^(/l+/2+ . . . +/„) + (/ili+/2f>+ •

Therefore {f,i,+f,l,+ . . . ^hL)-^

The expression {!iXi+f^x.,+ . . . -\-fnXn) is called
moment of the distribution referred to 0 as origin. We conclude that
when the chstribution is referred to G as origin, i.e. when deviations
aremeasured from the mean of the distribution, the first moment vanishes.

• • • +/n)

-L)ViIi^f2+ • • •

• • +/nln)]//l+/2 +

+/n)

r-fn)

. (1)

the first

Frequency Distribution Table.

(1) (2) (3)

Deviations of Var-
iables from some
fixed value.

Frequency of
Deviations.

Product of Nos.

in Col. (1) and

Col. (2).

Product of Nos,

in Col. (1) and

Col. (.3).

/i

/2

fz

K

fi^i

f^2

fn^n

f^2-

N

N'l

N'2

In the notation of the above table, where the dashes are omitted
in Nj, N, when the mean is origin, we have
;c=N'i/N andNi--0.

268 STATISTICS

Again, the root-mean-square deviation, s, measured from the
arbitrary origin 0, is given by

=N'2/N,
and N'2 is called the second moment of the distribution referred to 0
as origin.

Substituting as before we have

s''=Ul{^ + Lr^ ■ • • +/J•^•+^«)']/(/l+ ■ . • +fn)

jHf,+ . . ■ +/n) + 2^-(/i^i+ • • • +/n^J+(/lf\+ • ■ • -\-fnL')

(/l+ . • • +/J

since /i^it- • • • -^fJn=^-

Hence s''-x^-fcT^ . . . (2)

where a is the root-mean-square deviation measured from G as
origin, or the standard deviation as it is called.

From this result it is clear that ct is always less than s, or the root-
mean-square deviation is least when measured from the arithmetic
mean.

Generally, if we write

Vi:={h^'+ ■ • • +/.^/)/(/l+ • • • +fn),
V, = {fii^'--h . . . /nl/)/(/H- • • • +/n),

where 2J(fx'') and -S'i/I'") may be called the kth moments referred to 0
and to the mean as origins respectively, so that vi—O, V2=o^,
p'2=s^, we have

l''* = [/l(ll + ^f + • • • +/n(fn + ^)'']/(/l+ • • • +/n)

(/l+ • • • +/n)

For example, when h=2, since vq = ^ and 1^1 =0,

v^^v\-'k'' . . . (2) bis

Again, when A;=3, v3=-v' s—^i'i^—^ • • • (3)

and, when k=4, v^=^v\'-^i>3i-6v^x'-x* . . (4)

There are interesting statical analogues to the above results
concerning the mean and standard deviation.

APPENDIX 269

f

Let us imagine a set of weights, /], /g, fn . . . suspended at
Pj, Po, P3 . . . from a straight horizontal bar, and let the distance
of any typical weight / from some arbitrary origin 0 on the bar be x.
Then the first moment,

(where some of the x's may be negative corresponding to weights
suspended to the left of 0) measures the total turning effect of all
the given weights about 0, and if we further imagine all these

weights replaced by a single weight ^ -_-._- .v -^v

equal to their sum (/1+/2+ • • • ^ — "^—^ r^ —

-}-/„), then, in order to produce X

the same turning effect, it would /

have to be placed at a point G, the distance of which from 0
is given by

Thus x={f,X,^f,X,-i- . . . -^f^xJ/if,^f,-{- . . . +/J,

and, statically, this defines the position of the centre of gravity of
the given weights, /j, /g, •••/„, relative to 0.

As before, x=i:j{x-\-^)/i:f

=^x+SmEf;
hence f 1^1+^2+ ■ • • +AI«=0,

and, statically, this means that the turning effect of f^, fz • • • fn
about G is zero, in other words, the bar would balance freely about G.
Again, the second moment,

/l^"l+/2^'"'2+ • • • +/n^'n"j

measures the moment of inertia of the weights /j, /a • • • /« about 0,
and, if we imagine these different weights replaced by a single
weight (/i+/2+ • • • +/n) as before, the moment of inertia will
be unaltered if the latter be located at a distance s from 0, where

(/1+/2+ . . • +/nK^=(/r'«^+/2^^+ • • • +/„^V-);

therefore s'Mh^\+ . . . +Aa:„2)/(/,+ . . . _,_yj

=X'-\-a-,

as before, and the interpretation of this is that the square of the
radius of gyration of the system of weights about 0 equals the
square of the radius of gyration about G, the centre of gravity of
the system, together with the square of the distance of G from 0.
Also, 5 is clearly least when it is measured from G.

270 STATISTICS

G. The Mean Deviation a Minimum when measured from the
Median. Consider first the case when only two different values of
the variable are observed, X^, X2, and let their deviations from an
arbitrary value, 0, chosen as origin, be respectively x^, x^.

If /i, /g be the observed frequencies of these values, the sum of
their deviations from 0 is

^~ V X '^^' ^x ^'^^ich is clearly less when the

' t —^ — — — — y value 0 lies between X^, X2

' - than when it is smaller or

X, .r, o x^ Xj greater than both of them.

f^ f Choosing 0, therefore, be-

greater frequency we write the deviation sum

where x is the deviation of either of the values X^, Xj from the
other, and (fi—f.^) is positive since /i>/2.

Now this is evidently least w^hen {f-^— J 2)^1 vanishes, i.e. when
(1) ^1=0, in which case 0 coincides with Xj, the more frequent of
the two variables, or, when (2) /i=/2, and in this case, when the
two observed values occur equally often, the deviation sum is
constant for any origin between X^ and Xg.

When several different values of the variable are observed, they
may be arranged in order of magnitude, X^, X2, Xg . . . X„, from
the least to the greatest, with frequencies f-^, f.^, f^ . . . fn-

If /i>/„ we pair off /„ of the X„'s with /„ of the X^'s ; the devia-
tion sum for this pair is least and remains constant when measured
from any origin between Xj and x X X X«-i X

X„. We next pair off some or all i ^ 1 r' "i

of the Xj's which remain against ' " ''

an equal number of X^.^'s and the deviation sum for this pair is
least and remains constant when measured from any origin between
Xi and X„_].. If some X^'s still remain, we pair them off so far
as we can against an equal number of X^.g's but, if it be X„_i's
that remain, we pair them off against an equal number of Xo's.

This process can evidently be continued until ultimately we
reach the origin from which the mean deviation of the whole
distribution is a minimum, for if any X be left unpaired the origin
will coincide with that X. Otherwise, the deviation is least wheii

' APPENDIX 271

measured from any value between the last two X's paired ofE
together, and within that range it is constant.

Since, by definition, the median is the value of the variable half-
Avay along the series of given observations, ranged in order of their
magnitude and assigning each its due weight or frequency, it is
clearly such that a balance can be effected hy pairing off the values
on either side of it against one another in the manner explained
above ; it therefore follows that the mean deviation of a frequency
distribution is a minimum when the deviations are measured from
the median.

The statical analogy to the median also is worth noting. With
the same notation as before, the moment or turning effect of two
forces, /j, /a, about 0 is _^^_ . . ^ ^

/l^l+/2^2*

OX, X

But in this case, if 0 be taken y- \

at some point in between X^ y A

and Xg, since the mean devia

tion sums the separate devia- ^

tions Avithout regard to sign,

I

we must imagine /^ reversed ^

so as to produce a turning effect in the same direction as before.
The moment \\d\l then be still (/la-'j+A^'a)) ^^'^ i^ is less when 0
occupies such a position than when it is on X^Xg produced in
either direction.

Taking 0, therefore, somewhere in between X^ and X2, the moment
may be Avritten

=/2(^i+-'^'2)+^i(/i-/2) ;

and, if /i>/2, this is least when x^ vanishes, that is, when 0 coincides
with Xi, but if /i=/2, the two forces constitute a couple, and the
moment is the same whatever position 0 occupies between X^
and X..

7. The Method of Least Squares. To the student \\'ho is un-
acquainted with the dififerential calculus, the following descriptive
argument, the basis of the principle of least squares, for determining
the values of m and c which make

(ma;i+c-2/i)- + (m.r2+c-y2)-+ • • • -\-{mx„-^c—y„)- ... (1)

a minimum, may prove instructive.

Let us call the above expression E and let us suppose that different
values are given to m while c remains unchanged ; in that case E

272

STATISTICS

will vary with m, and we might imagine the different values obtained
for E plotted against the corresponding values of m giving a curve
of some type. Such a curve may rise and fall in wave -like fashion
as in the figure, resulting in maximum points like A and C, and
minimum points like B, where we define a maximum point to be
such that, as we move away from it along the curve, whether to
left or right, the size of the ordinate (and therefore the value of E)
decreases ; likewise, a minimum point is such that, as we move
away from it, the ordinate (and therefore also E) increases. In
the neighbourhood of such points it is clear that the size of the

ordinate, such as Aa or B6,
changes so slowly as to be
practically stationary.

Suppose then that m and
(m-j-ju.), />t being very small,
are two values of m respec-
tively at and near a minimum
position on the curve, i.e. a
position like B corresponding
to a minimum value for E.
Since E near such a point
does not differ appreciably from E at such a point, we may prac-
tically equate the two expressions obtained for E by substituting
(m+/i) and m respectively for m in (1), thus

(m + yLi.Ti + C-i/i)2-f (wi+/X.T2 + C-2/2)2 +

= (m.ri+c-?/i)2+(mx'2+c-?/2)2+ . . .

=(mXi+c-?/i)2+(ma:2+c-2/2)--f- • . .

[(maJi+c-i/i)2+2/xa:j(m.Ti+c— ?/i)+/^Vi]+ . . .
=(wa;i+c-?/i)2+ . . .

Thus [2xi{mXi-\-c—yj)-\-ixx'^j}-\- . . . =0.

Now, the smaller we take fx, the nearer to the truth does this
result become. Hence, by making fx tend to zero, we are led to
the strictly true relation

Xi{mXi^c-yi)-{- ... =0.

This is one of the equations in the text. To obtain the second,
we keep m constant and vary c.

Suppose c and (0+7) are two values of c at and near a minimum

APPENDIX 273

position on the curve ; then, equating the two corresponding
values of E, we have as before

(m^i+c+y— yi)-+

={mx\+c—yi)- +

-={mx\ + c-yi)--{-
=0.

[{mx^-\-c-yi)- + 2y{mXi+c-yi)+y-]+
Thus [2(wx-i+c— ?/i)+7] +

and, proceeding to the Hmit when 7 tends to zero, we reach the
other equation in the text, namely,

(m.ri + c-y,)+ . . . =0.

[The Method of Least Squares came first into prominence in
Astronomy in connection Avith the determination of the best value
to take when a number of observations, apparently equally reliable,
give results not quite in agreement. If, for instance, x be the true
value of some variable, and if .rj, x.,, x^ . . . a'„ be the results of
n observations, the method of least squares assumes x to be given
by making

ij = (^x-Xi)'^ + {x-x.;,)--\- . . . +{X-Xnf
a minimum.

Now — =2(,f— a;i)-r2(.i-— .{•„) |- . . . +2(.f—.T„), and this vanishes
dx

Avhen {x—Xi) + {x—x.,)-\- . . . +{x—xj^0,

i.e. a;=(.ri4-.r2+ . . . -j-xj/n,

so that in this case we are led to the ordinary arithmetic mean of
the 71 observations as the best value.

The method was used by Gauss as early as 1795.]

8. To prove

•+00

J -co

^''Ax^Vtt.

Let

.'-oc

thus, also,

r+co

1= e-^'dy;

.'-co

therefore,

12= e-^dx

+03

' -co
'+00 ,"+co

r+oo ,-+00
= I I e-^^+'^'^dxdy

J -OD .' - CO

= e-'ydrdd

Jr== o.'e=o

s

274 . STATISTICS

(by changing to polar co-ordinates)

.'o .'0

=

=(i)(27r).

Hence 1=1 e-''''dx = Vn.

J -OS

9. To prove :—

(1) r(n+l)=nr(n). (2) B(m, n)=

(1) r(?i+l)=i a;"e-^dc

.'o

,-co

= — / .x-"rf(e-*)

.'.r--n

r(m)r(n)
r(m+n) '

.r"e-
=nT{n),
because the expression in square brackets vanishes at both limits.

(2) r{m)r{n)=f e-^^"'-hIi( e-^rj^'-^dr,

Jo Jo

/■CO I'CO

— I e-^'^x-"^'-'2xdx\ e--''~ij-'^--2ydy,
Jo Jo

where x-=^, y'^=^r].

•cm rco

Hence r{m)r{n)=i e-^'^+-'^'>x-"'-hj-'*-Mxdy

.0 .'o

= / e-'Y-"'+2«-2 cos2"»-'0 sin^"-!^ rdrdO

Jr = oJe=--o

(by changing to polar co-ordinates).

Thus r{m)r{n)=! e-'V-'"+-"-Wrf cos-'^-W sm-^-^ddO

Jo .''J

if

g-;'^m+n -Ig^p

- 1^-1(1 -^)'""'rf^l.

where p=r2 and /:=sin2^ ;

therefore, r(m)r(w)=P(m+?t)B(w, m)

^r(m+w)B(w, w)
by symmetry.

APPENDIX

276

10. Elementary Method of Testing the Probability Integral Table.

The reader may find more satisfaction in using the probability
integral table if he tests for himself one or tAvo of its results by
means of squared paper or in some other wixy.

We have seen that the probability of an error between 0 and a^
is given by the expression

V27r.'o ^

Put ^ — V2x, and this becomes
1

- — i e-'^dx^i e-^dx \

e-^W.r, by Note (8)
=area OBPN/area A'BA, in the tigure.

Now the graph of y=e~'^ is draAvn in fig. (40) of the text, and it
is possible therefore to get an
approximation to the above
result for any value of x by
counting the number of small
squares in that figure enclosed
by the areas corresponding to
OBPN and A'BA respectively.
Each complete small square
may be reckoned as 1 , and each
portion of a square may be
reckoned as 1 if it exceeds half a square and as zero if it is less
than half a square.

This gives, for example,

1 rO-25

*^V.r =98/707 =0-139,

jy=e-

Vtt-'O

whereas the tables give 0-138.

For a value like ;r=0-71, count the squares in the usual way
between curve, axes, and ordinate a;— 0-70 : then add to the result
one-fifth of the number of squares in the small slice of area between
curve, axis, and ordinates .t;=0-70 and x=0-~o. We get

1 /■""' o

— = e-^c;a;=240/707 =0-339

as compared with 0-342 from the tables.

These results are not unsatisfactory considering the rough nature
of the method followed to obtain them.

276 STATISTICS

11. The Law of Fretiuency in the case o! two Correlated
Variables with certain Deductions therefrom— [based on Professor
Karl Pearson's memoir, Regression, Heredity and Panmixia {Phil.
Trans., vol. 187a, pp. 253-318)].

Consider two variables whose deviations, x and y, from their
resiJective means are due to a number of independent causes, the
deviations in which from their means can be quantitatively denoted
by €i, €2, • . . e™.

We assume that each e deviation is so small compared to the
mean value from which it is measured that x and y can be sensibly
expressed as linear functions, thus

a;=aiei+a2e2+ • • . +a»ie,„ • • • (1)

2/=6iei+V2+ • • • -^K^n ' • • (2)

(Some of the as and 6's may be zero, and if x only involved, say,
fj, €2 ■ • • €;., and y only involved ej.+i . . . e,,^, then it would be
natural to expect no correlation between x and y.)

We further assume that each e varies according to the normal
laAV with S.D. CT with appropriate suffix.

Equations (1) and (2) show that the same x and y may arise in a
multitude of different ways obtained by varying the e"s so that
their weighted sums (the a's and 6's being the weights) remain
unaltered. The probability that the particular deviations l^'ing
between

ti„(ei+Sfi)> 62.^(^2+8^2)' • • • e„„(e,„ + Se,J
shall concur, since they are all independent, is

\GiV27r / \a„y27T

But, \vriting

a3e3+ • • • +ct„,e„,=a, 6363+ . . . +6„,e,^=^,
equations (1) and (2) become

aiei+«2f2+(a— •^■)-=0

^e,+6262+(/3-y)=0.

Therefore -

And, for any function v,

J J J J \oei de-2 oe^ oej/

=r (a^b^- a2bi)jjvdeid€2.

APPENDIX 277

Hence

The ^o/aZ ijrobability for deviations between a; (x-j-Sx) and
y^iy+Sy) is obtained by integrating z between limits — CD and +oo
for all the e's from e^ to e,,,, and it is not very difficult to see that
this will ultimately lead to an expression of the form
C . SxSy . e-^^^-'^^^-y+^y''\

This is the required law of frequency.

To find the meanings of the constants a, b, h. The total probability
for a deviation between .r (.r+Sa;) associated with any deviation y is

.'-co
J -co

^CVTT/bSxe-'''^"^-''-^'\

But if X be subject to the normal law, the probability for a devia-
tion between a;^(.r+8.r) is
^ 8x

-a-'/io-x^

V277 . a,

where a^. is the S.D. of x independent of y.

Comparing these two results, we have

l/2a^'=(ab-h')/b=a(l-r='),
if r = — h/Vab.

Similarly, l/2CT/=(ab-h')/a=b(l-r'),

so that h=— r\/ab=— r/2a^o-y(l— r").

Again, we may integrate z for all values of x and y, and so get
the total frequency, N, of the {x, y) pair.

,--rCO -+00

Thus. N=C e-^'-'''-^-^'y+''^-^dxdy

. -co . - CO

=CV7Tjbj'^\-''<"^-^"-^'''dx

J -co . ■■

=^CVT^bVWibl{ab-¥)l

278

STATISTICS

Hence

TT

TT

N

27ra,(T,V(l-^')
Thug 2=Ce 2(i-7-.^)L<r/^ cr.^, <-JS.rSi/,

where C has the above value.

It still remains to interpret r and to see that it is really the
coefficient of correlation as defined in Chapter x. For this purpose
let us suppose we have observed w pairs of associated x's and y's,
namely

The probability for such a concurrence, taken along with a given
value for r and assuming the observations independent, is pro-
portional to

1 1 rSt2 2>-2a\v 2j^|"l

_ ^^ [-in-^rnK],

= {l-r-)-"^-e 2(1-'-)
where k ^Uxy/nar^y

-^ g 2 1 - /■-

Now the probability of this particular distribution is greatest

when -

i log {l-r^-) + -^
l—r-

is least, and, differentiating with respect to r, this leads to

2r {l-r^-){-K)-\-2ril- Kr)
*. -i- u

h-f^ (l-r-)2

i.e. -r(l-r-)-/c(l-r-) + 2r(l-«?-)=0,

i.e. -r+r3-/<:+/cr2+2r-2/fr2=0,

i.e. (r-«)(l + r2)=0.

It is not difficult to show that r=-K gives a minimum ; hence the
required probability is a maximum and we get the best value for
the coefficient by taking

r —/f =Z'xy WjCTy.

APPENDIX 279

CERTAIN CURRENT SOURCES OF SOCIAL STATISTICS

Any one who is anxious to get reliable figures bearing upon some
social matter is somewhat at a pause unless he is thoroughly con-
versant with all the statistical ramifications of Government autho-
rities, local and national, of trade unions, friendly societies, and
hosts of other bodies of a public or semi-public character.

While recognizing the lavish outpouring of statistics of all kinds
upon a multitude of diverse topics every j'ear, and aitpreciating the
immense care and patience shown by those who are responsible for
their collection and preparation, one cannot but deplore the lack
of any co-ordinating principle in general between one body and
another either in deciding what statistics shall be collected, by
whom and when they shall be collected, or how afterwards they
shall be tabulated and presented to the public. Too often a narrov.-
minded jealousy prevents one authority from consulting mth
another, and such co-operation as does exist is due largely to the
efforts of able and enlightened individuals. The result is that a
vast amount of labour and expense goes waste and the loss to the
public is incalculable, but the public do not care, and they do not
care because they do not know.

At present, to quote from an influential petition on the subject
recently presented to His Majesty's Government, ' It is almost
universally the case that any serious investigation is reduced to
roughly approximate estimates in relation to some factor which is
essential for its result. ... It is not too much to say that there is
hardly any reform, financial, social, or commercial, for which adequate
information can be provided with our present machinery.' But
this state of things Mould be partly remedied by adequate control
such as might be secured by the establishment of a central statis-
tical office M-ith a minister in charge who should be responsible for
unification so far as possible in the collection, tabulation, and issue
of all public statistics.

It is scarcely possible for a single private individual to make
a quantitative investigation of any social question on a large enough
scale to produce results of real value ; conspicuous instances like
Booth and Rowntree might seem to be exceptions to this rule, but
even they had a number of workers acting under their direction,
without whose aid their task would have seemed almost hopeless.

280

STATISTICS

For such statistics as we have we are therefore dependent upon
Government departments, local authorities, public officials, trade
associations representing employers or labour, public companies,
and so on. The reader Avho A\dshes to get some idea of the extent
and the limitations of official British statistics is referred to the
admirable introductory chapters of Bowley's Elements of Statistics.
Here we cannot do more than mention a very few of the most
important sources whence such statistics are derived.

The most voluminous of all our records is probably the Census
of the Pojyidation which is taken every ten years. Its scope is but
faintly realized by enumerating the chief subjects on which the
Registrar- General asked information from each householder in 1911,
namelj' :

(1) Numbers and Geographical Distribution of the Population.

(2) Nationality' and Birth-place.

(3) Numbers at Different Ages, Male and Female.

(4) Numbers Single, Married, and Widowed.

(5) Sizes of Families, including Children Dead.

(6) Numbers engaged in different Professions and Occupations.

(7) Numbers Blind, Deaf, Dumb, not in their Right ]\Iind.

(8) Numbers occupying Dwellings of Different Sizes as measured
bj' the Number of Rooms.

This may seem an ambitious scheme when it is stated that the
mere enumeration of the people was successfully opposed less than
two hundred years ago as ' subversive of the last remains of English
liberty and likeh^ to result in some public misfortune or an epidemi-
cal disorder,' and the first census was only taken in 1801. [See
Article in the Encyclopaedia Britannica on the subject.]

The results of each census are published in bulk}' volumes as
soon as they can be reduced and tabulated, a process which, of
course, takes a considerable time even for an army of workers
with calculating machines and every modern device to facilitate
their progress. It is to be regretted that more is not done to
advertise so valuable a record of work by publication in a cheap
and attractive form of a summary of matters which vitally affect
the good of the commonwealth. As it is, the census volumes tend
to be purchased only by public authorities and officials who require
to use them occasionall}' as books of reference.

Neglect of the blandishments of advertisement— to be commended
in general because such neglect is somehow associated with the
presentation of all truth^may be perhaps carried too far in the
issue of statistics.

APPENDIX 281

It will be noted that in the periodical census no mention is made
of wages though the people are classified as regards occupation,
and for information upon this point we must turn to another source.
The last general census of wages was taken in 1906, following
and improving upon an earlier inquiry twenty years before, but,
in connection with an inf(uir3^ by the Board of Trade into the cost
of living of the working classes, information was collected as to
rates of wages in 1912 of Avorkpeople in certain occupations in the
building, engineering, and printing trades, these being selected as
industries common to most towns, and because the time rates of
^\■ages paid in them are largely standardized.

The 1906 inquiry into earnings and hours of labour, unlike the
decennial census, was conducted on a voluntary basis and was
never wholly completed. In brief it set out to discover from
emploj'ers : —

(1) The Numbers of Working-people Employed in Various
Occupations, distinguishing Men, Women, Lads, and Girls.

(2) The Nature of the Work done and the Rates of Wages Paid,
distinguishing Time Rates from Piece Rates.

(3) The Hours Worked, distinguishing Under- or Over-time from
Normal Time.

The ground actually covered by the inquiry embraces the fol-
lowing trades : Textiles, Clothing, Building and Woodworking, Public
Utility Services, ]\Ietal, Engineering, and Sliipbuilding — in 1906 ;
also Agriculture, and Railway Service — in 1907 ; the reports upon
these trades were published separately at different dates between
1909 and 1912, and the following trades were bulked together in
one volume, published in 1913 — Paper and Printing ; Potter}-,
Brick, Glass, and Chemicals ; Food, Drink, and Tobacco ; and
Miscellaneous Trades.

The Cost of Living Inquiry of 1912 was in continuation of a
similar inquir}- in 1905, wliich in addition compared conditions in
the United Kingdom and certain foreign countries. It dealt not
only with wages but also with rents and retail jyrices.

The report states that ' particulars as to the rent and accommo-
dation of typical working-class dwellings were obtained from
officials of local authorities, surveyors of taxes, house owners and
agents, and by house-to-house inquiry.' Also ' returns of the
prices most generally paid by working-class customers for a number
of specified commodities were obtained in each town by personal
inquiry from a number of retailers engaged in working-class trade.'

Since then Lord Sumner's Committee and a Committee of tha

282 STATISTICS

Agricultural Wages Board have examined the change in the cost of
living between 1914 and 1919, as evidenced by a number of house-
hold budgets collected from among urban working- classes and
workers in rural districts respectively.

One other highly important inquiry carried out by the Board of
Trade deserves notice, namely, the First Census of Production of the
United Kingdom (1907).

The published report shows : — «

(1) The total Net Output in Money Value for each Trade Group
in each Industry.

(2) The Number of Persons Employed in each Trade Group
(salaried persons and wage-earners exclusive of outworkers).

(3) The Net Output per Person Employed in each Trade Group
as deduced from (1) and (2).

(4) The Horse-power of Engines in Mines, Quarries, or Factories
Employed in each Trade Group.

It is explained that the term ' net output ' here represents the
value of the aggregate output of the factories, etc., from which
returns were received in each trade group, after deducting the cost
of materials purchased from factories, etc., not included in the
group, or supplied by merchants or others not making returns to
the Census of Production Office.

Valuable as the results of these inquiries undoubtedly are, they
would be of still more value were it only possible satisfactorily to
collate the various returns of population, wages, and production.
No record of wages was included, for example, in the Census of
Production statistics, and it is quite impossible to deduce the number
of wage-earners and those dependent upon them in any trade at
any given time.

Apart, however, from such special inquiries as we have instanced,
and the ten-yearl}^ census of the people, there are other periodical
records issued Avhich jarovide us with valuable information. The
Ministry of Labour, until recently a special branch of the Board
of Trade, charged Avith the duty of keeping in touch with labour
.conditions, issues each month a Labour Gazette giving particulars
relating to the state of employment in the principal trades in the
United Kingdom based on returns from employers, trade unions,
and employment exchanges, besides information concerning trade
disputes, changes in wages and hours, the course of prices, railw.
traffic receipts, foreign trade, etc. The Board of Trade also jsub-
lishes M-eekly a Journal and Commercial Gazette dealing A\'ith matters
of interest to all who are engaged in commerce or finance ; while a

APPENDIX 283

Monthly Bulletin of Statistics of production, trade, finance, employ-
ment, etc., at present issued under the name of the Supreme
Economic Council, is an important recent addition to our knowledge
of international statistics.

Again the Registrar -General makes a quarterly return and annual
summary of births, marriages, and deaths in the different counties
of England and Wales, and of births, deaths, and infectious diseases
in certain large towTis. In each public health area the medical officer
reports periodically upon the hygienic condition of the district and
the health of the people under his care. The Board of Education
is answerable for conditions in the schools, and the Home Office
in factories and prisons ; they report from time to time. The
Ministry of Health similarly issues returns relating to pauperism
and to housing, while the Board of Agriculture and Fisheries registers
the acreage under crops and the number of live stock in the United
Kingdom, and the Commissioners of Customs record the expansion
or contraction of foreign trade.

In addition we have the endless accounts and statistics supplied,
some voluntarily and some compulsorily, by municipal bodies,
l^ublic companies, banks, trade associations, co-operative societies,
insurance companies, trade unions, etc.

And yet, in spite of all this wealth of statistics, some surprising
gaj)s occur, as we have already seen, in important particulars
which cannot be traced. We shall quote only one more instance
of such a hiatus — the income-tax returns provide a basis for measur-
ing that part of the national income which is subject to taxation,
some idea also can be formed of what the wage-earners receive,
but as to the earnings of the portion of the community falling in
between these two classes we are entirely' ignoi'ant. It is possible
that war conditions during the years 1914-19 may have vastly
increased the knowledge of the Government as to some matters
such as internal resources and inland trade, of which little Mas'
loiown before, but, if so, the public, ^hom it concerns so closely,
have not yet been permitted fully to share in this advantage.

For an excellent summary of labour statistics compiled or col-
lected by the Government the reader is recommended to consult
the Annual Abstract of Labour Statistics of the United Kingdom,
Dublished in the past by the Labour Department of the Board of
rade.

Note. — A most useful Guide to Official Stati»(ics is now issued l»y H.M.iS.O.
Dr. Bowley's Official Statistics will repay careful study in conjunction with it.

284

STATISTICS

A NOTE ON TABLES TO AID CALCULATION

The short tables which follow are only inserted as specimens, as
it is expected that the reader who wishes to make extensive use
of such tables will have access to the fuller ones to which reference
is made below.

2-00 2-50

Probability Integral Table, giving area of curve z = ~~--=^e *^'' in

V27T

terms of corresponding abscissa, see fig. (55) : —

i

Kl + a)

a

1

ill + a)

a

•00

•50000

■00000

•76

•77637

•55274

•10

■53983

■07966

■77

•77935

•55870

•20

•57926

•15852

•78

■78230

•56460

•30

•61791

■23582

•79

■78524

•57048

•40

•65542

■31084

•80

■78814

•57628

•45

•67364

■34728

■85

•80234

•60468

•50

•69146

■38292

■90

•81594

■63188

•55

•70884

■41768

■95

■82894

•65788

•60

•72575

■45150

100

■84134

•68268

•65

•74215

■48430

105

•85314

•70628

•70

•75804

■51608

110

•86433

•72866

•71

•76115

■52230

150

•93319

•86638

•72

•76424

■52848

200

•97725

■95450

•73

•76730

■53460

250

•99379

•98758

•74

■77035

•54070

300

•99865

•99730

•75

•77337

■54674 !

1

3-50

•99977

•99954

Fig. (56), the result of plotting a against |, enables us to estimate
the probability of an error tying between any two limits.

APPENDIX

285

Table giving P, to test ' goodness of fit,' corresponding to certain
v.alues of n' and yj : —

v'

x-->i

5 C

'

8

9

1<»

11 12

13

1^
02964

15
•02026

•<;7Ht)8

•54381 i •42319 ' -32085

-23810

•17358

•12465

•08838 06197

•04304

S

•7707S

•OoniXi -53975 ^42888

-33259

•2520(i

•18857

•13862 10050

-07211

•05118

•03600

;»

•s:.7i2

•757-58 ' -0472:5 ^SfJOo

•4:'.347

•34230

•20503

•20171' 1512(1

-11185

(18170

•(J5914

in

•iHUl

•834311-73992 •(53712

-53415

•43727

•35048

•27571 21331

-10201

.122:52

•09094

11

•1)4755

•891181 •81520 ^72544

•02884

•53210

•44049

•35752 i ^ajOO

-22307

■17299

-1:J2(J«>

Vl

•9()!);i2

•93117 -87330 \ -79907

•71330

•62189

•53039

•44326 -36204

•293:53

•23299

•182.50

X.\

•!)8344

•95798 [•91008 '•85701

•78513

•70293

•(;1590

-52892

-445(J8

•:?0904

•30(171

-24144

\\

•!»;ni!t

■97519-94015 -90215

•84300

•77294

•09:;93

-01082

-52704

-44781

•37384

-:'.0735

15

•!)9547

-98581 ' -90049 i -93471

•88933

•83105

•70218

-08604

•60030

-52(152

•44971

-37815

One of the earliest tables of the probability integral appeared in
Kramp's Analyse des Refractions (Strasbourg, 1798), where the
calculation of J"e~^-<Z.T was given to eight places from a;=0 to a; =3
at intervals of 0-01. Tables more recent and extensive are those
due to J. Burgess [Trans. Roy. Soc. Edin. 1900) and to W. F.
Sheppard {Biometrika, vol. ii.,.pp. 174-190), Of these the latter

"■ 1 1 1 M i i i III "t^l \ \ 0 \

iJ_ J . JJ- .L X L X u/ J £L - ' :f- 1

Graph }}f rhe^cirLe a- -7= j/|-e 1-f c ^

! ! ' ^-. •■ ■* *

^^^'•*'''*^

' ^^ j

1 ^:i^0^ 1 "^

^

^^ J_ 1

^^

'^^ ^^ 1 , ^ ui ""

^ ^

^

>'' . _' __ _ _ ±

«^_ - _

J' - i

/ X

Jl'X-

r< ' 1 _._ L^

1-50

IMU. (50).

2-00

2-50

is reproduced in the admirable Tables for Statisticians and Bio-
metricians, edited by Karl Pearson (Camb. Univ. Press, 1914), and
the same volume also contains Palin Elderton's P Tables for testing
' goodness of fit ' which first appeared in Biometrika, vol. i., and
Duffell's Tables of the Logarithms of the T Function from Biometrika,
vol. vii., besides a large number of other valuable tables.

It should be remarked in connection with the last-named table that
the formula P(.r-j-l)=.r T{x) enables us to reduce the calculation
of any T function to one in which x lies between 1 and 2, by repeated
applications of the logarithmic relation, thus
logr(x-+l)=log.r+logr(x)

=log .r+log (a;--l) + log r(a;-l),

286

STATISTICS

and so on. When x is large, however, say greater than 10, the
well-known approximate formula

(see, for instance, VVhittakers Analysis, § 110) will be found useful,
and it may also be WTitten

1 r(a;+l) n QoonQoo , 0-03619121 , . ,
log — ^ ^=0-3990899-f +* log x,

a form often convenient.

It may be of service to record here the values of a few constants
which frequently recur for speedy reference :

,

c = 2-718 2818

jr = 3I41 5926

logio 2=0-301 0300

1 = 0-367 8794

c

Iogio7r = 0-497 1499

login 3=0-477 1213

logio 6=0-434 2945
logio(log,oe) = 1-637 7843

logio ^==1-600 9101

V27r

The statistician who has Pearson's Tables, Barlow's Tables of
Squares, etc., together with a good set of Tables of Logarithms
(unless he is so fortunate as to have a mechanical calculator, for
instance a Brunsviga, at his disposal) and of Trigonometrical
Functions such as Chambers's Seven-Figure Tables, may consider
himself amply provided for serious research and decidedly better
off than his predecessors aaIio prepared the way for him by doing
great work with much poorer tools.

MISCELLANEOUS EXAMPLES

[Selected from Lomion B.Sc. {Eeon.) Pass and Honours papers']

PART I

(1) Define the genus ' average,' and the principal species of that
genus. Adduce concrete cases in which (a) the Arithmetic Mean, or
(6) the Median, is specially appropriate.

(2) Supposing that statistics of rents of working-class dwellings
have been collected in a certain district for a seines of years, describe
some way of forming an index number showing the changes in rents
from year to year during the period. Give reasons for the process
you adopt, or state any advantages it appears to you to possess.

(3) Measure by whatever method you think most suitable the
correlation between the two following series, and show graphically the
relationship between the two series.

Exports

Unemployment

Exports

Unemployment

per head.

Index,

per head.

Index.

£

£

1884

6-5

8-1

1899

(1-3

2-2

5

5-9

9-3

1900

7-1

2-5

6

5-9

10-2

1

6-7

3-3

7

6-1

7-6

2

6-8

4-0

8

6-4

4-6

3

6-9

4-7

n

6-7

2-1

4

71

6-0

1890

7-0

2-1

5

7*7

5-0

1

6-5

3-5

6

8-7

3-6

o

6-0

6-3

7

9-7

3-7

3

5-7

I'O

8

8-5

7-8

4

5-6

6-9

9

8-5

7-7

5

5-8

5-8

1910

96

4-7

6

6-1

3-4

1

10-0

30

/

5-9

3-5

o

10-7

3-2

8

5-8

2-9

3

11-4

2-1

(4) Apply some test by which the figures in the previous table can
be used to determine whether unemployment (as there measured)
increased or diminished in the 30 years.

(5) Exliibit the difficulty of comparing nations, in respect of poA\er
and prosperity, by means of statistics relating to [n) the number of
population, (6) occupations, (c) criminality, {d) exports and imports.

T

288

StAtlStlCS

(6) Draw up, with careful attention to form and detail and showing
all sub-totals, a blank table in which could be shoA\-n, for the years
1919 to 1923 inclusive, the numbers of students Avho entered for the
Final Examination for B.Sc. (Econ.), distinguishing Internal and
External Students, Pass and Honours Candidates, and the results of
the examinations (Pass or Fail in the case of Pass Candidates and
Honours I, II, III, Pass Degree. Fail in the case of Honours
Candidates).

(7) Define the geometric mean, and discuss its use in forming index
numbers of prices.

(8) The average prices of wheat and the quantities sold at four
markets are given as follows : —

Market.

Average Price per Qr.

Quantity sold, Qrs.

A
B
C
D

27s. 3d.
28s. 8d.
29s. Id.
279. 2d.

36,000

1,000

16,000

12,000

Find the mean price for the four markets, weighting each local
average with the quantity sold.

Would it be possible for the average price at each of the above
markets to rise from one year to the next and yet for the weighted
mean price to fall ? If so, under what conditions ?

(9) Illustrate the necessity for standardisation when hetero-
geneous groups are in question by describing the methods of comput-
ing standard bii'th- or death-rates or family food-consumption.

(10) The following are the Annual Premiums required to secure
at death £1000 plus a Guaranteed Reversionary Bonus of £2 per cent,
on the sum assured under the Whole Life Policies of a certain
Assurance Company : —

Age next
Birthday.

Annual Premium.

25
30
35
40
45

£ s. d.
24 12 6
27 14 2
31 11 8
36 7 6
42 6 8

Find by any method of interpolation what the jH-emium would be
at age 36 next birthday.

MISCELLANEOUS EXAMPLES

2S9

(11) Explain why the method of measuring the mortality from any
disease by the proportion of deaths from that disease to deaths from
all causes is essentially fallacious.

Criticise the following mode of argument in a recent blue-book,
containing anthropometric data with respect to school children :
The gradation in weight from the poorest group up to the wealthiest
is one of the most striking features of the tables. If A\e take all the
children of ages from 5 to 18, we find that the average weight of
the boy from a one-roomed tenement is 52-6 lb. ; of the boy from a
two-roomed tenement, oG-l lb. ; of the boy from a three-roomed
tenement, 60-6 lb. ; of the bov from a tenement of four rooms or more,
()4-3 lb.

(12) Show how to measure the ' trend ' and the ' fluctuation ' of a
series of numbers relating to economic phenomena, such as trade or
employment.

(13) Find the average age and the median age of the married men
included in the table below, and calculate one measure of dispersion.

Ages.

Maiiied ."Men.

Widowers.

NuRiber of Men

Average Age of Children

Number of Men

OOO'a.

under 10.

OOO's.

Under 20

I

■47

20—

34

•61

25—

99

•97

1

30—

132

W,0

o

35-

139

1^99

3

40

138

1-98

5

45

130

b53

7

50—

104

•95

9

55—

78

•48

11

60—

53

•20

13

65-

33

•09

13-5

70—

15

•06

10^5

75—

6

•04

7

80—

2

•04

4

(14) What do you understand by a weighted average ? Estimate
the average number of children of married men of all ages from the
data in the above table.

(15) Estimate the number of married men between the ages
52 and 53 in the same table, and also estimate at what age the
average number of children is a maximum. Illustrate each estimate
by a diagram.

(16) Define frequency group and standard deviation. Show that,
if m'2 is the second moment of any frequency group about any

290

STATISTICS

origin and rn., the second moment about the average of the group,
and X is the average measured from the origin, then m^^^m'^^x^-
Calculate the standard deviation of the ages of widowers sho^\^l in the
table of question (13).

(17) State the product-sum formula for the correlation coefficient,
and prove that if the means of rows and of columns of the correlation
table lie on two straight lines the equations to these lines are

'.(/.

ij=zr~^x,

respectively, x and y being deviations of the variables from their
arithmetic means, r the correlation coefficient, and crj, o-y the
standard deviations.

(18) Below are given the populations of the County of London and
the four surrounding administrative counties at the Censuses of 1891
and of 1901 :—

Count}-.

Population.

1891.

1901.

London

Essex ....

Middlesex .

Surrey

Kent ....

Total .

4.228,317
578,471
542.894
419,115
807,328

4.536,541
816,640
792,314
519,654
936,240

6,576,125

7,601,389

(a) Assuming a constant percentage-rate of increase in each
administrative county, estimate its population in 1896 at a date
midway between the two Censuses.

(6) Assuming a constant percentage-rate of increase for the area as
a whole (London and the four surrounding counties), estimate the
total poi:)ulation at the same date in 1896.

Why does your estimate (6) differ from the sum of the estimates
under {a) ?

(19) Give as exact a definition as possible of the term ' Cost of
Living.' How far can the change in the Cost of Living be measured
over a period in which there have been considerable modifications of
diet or other changes in consumption of necessary commodities ?

(20) Discuss the methods of presenting A\age statistics by averages.
Illustrate by a diagram the following data : —

Building Trades. — Men. Full time earnings. Median, 37s. ;
Quartiles, 29s. 6d., 40s. 6d. ; 5-9 per cent, received less than 20s. ;
2-8 per cent, received 45s. or more.

Estimate the average wage roughly.

MISCELLANEOUS EXAMPLES

291

(21) Construct a diagram to show graphically the relationship
between yield of corn and rainfall from the data in the table below.

Years.

Yield per acre

Rainfall in

Years.

Yield per acre

Rainfall in
July, in inches.

of corn, in bushels.

July, in inches.

of corn, in bushels.

1886

24-5

Mo

1896

40-5

617

7

19-2

2-40

7

32-5

3-59

8

35-7

3-83

8

300

2-84

9

32-3

4-45

9

360

3-42

1890

26-2

203

1900

37 0

415

1

33-5

1-88

1

21-4

2-63

o

26-2

3-71

2

38-7

4-78

3

25-7

2-20

3

32-2

341

4

28-8

1-58

4

36-5

5-23

5

37-4

601

5

39-8

4-78

(22) Find the correlation between peld of corn and rainfall in the
above table.

(23) Define (a) arithmetic average, (6) geometric average, (c)
median, (d) mode, {e) quartile.

Instance cases when (6), (c) and ((/) are speciallj^ appropriate.

(24) Comment on the form of grouping adopted in (26) Table I.,
and state any inconveniences that it presents.

Calculate approximately the values of the median and quartiles,
using a graphic method.

(25) Explain what is meant by the skewness of a frequency dis-
tribution. Give Pearson's measure of the skewness, and any other
way of measuring it.

Obtain some measure of the skewness of the distribution in
(26) Table II.

(26) Table I., showing the number of civil parishes in England and
Wales in which the population at the Census of 1901 lay between the
limits given in the column on the left : —

Population.

Number of
Civil Parishes.

Population.

Number of
Civil Parishes.

1,557

842
2,411

413

241

273

None

1 and under 50
50 „ 100
100 „ 200
200 „ 300
300 „ 400
400 „ 500

25
812
1,339
2,503
2.036
1,410
1,038

500 and under 750

750 „ 1,000

1,000 „ 5,000

5,000 „ 10,000

10,000 „ 20,000

20,000 and upwards

Total No. of CivU Parishe.^

14,900

292

STATISTICS

Table II., showing the number of rooms measured, in a certain
investigation, in which the size lay between the limits given in the
column on the left ; area calculated to the nearest square foot : —

Write a short account of the use of graphic methods in statistics.
Draw diagrams representing the data of Table I. and of Table II.

(27) Define the standard deviation, and show that the mean square
deviation is least when deviations are measured from the aiithmetic
mean.

Fmd the mean and standard deviation for the sizes of the rooms
given in (26) Table II.

(28) What corrections are applied to the crude death-rates of
areas in order to obtain comparable rates ?

(29) (1) Estimated average weekly wages of agricultural labourers
in thirtj^-six counties of England in 1891. and (2) the percentage of
the population in receipt of poor law relief, in rural unions of the
same counties, on 1st January of the same year : —

Percentage
in Receipt
of Relief.

County.

\Yages.

s. d.

1

18 6

o

18 0

3

17 0

4

17 0

5

16 0

6

16 0

/

15 G

8

15 6

9

15 0

10

15 0

11

15 0

12

15 0

1-7
2-3

2-5
21
30
21
2-8
2-7
3-5
31
31

$>-7

County.

Wages.

s. a.

13

14 8

14

14 0

15

14 0

16

14 0

17

13 0

18

12 0

19

12 0

20

12 0

21

12 0

22

12 0

23

12 0

24

12 0

Percentage
in Receipt
of Relief.

County.

Waj.

es.

Percentagf>
in Receijit
of Relief.

s

a.

3-6

25

12

0

4-9

31

26

12

0

4-7

40

27

12

0

39

2-3

28

11

6

40

2-8

29

11

6

4-5

4-5

30

!1

6

4-2

4-7

31

11

0

5-2

4-7

32

11

0

4-2

5-7

33

10

6

4-2

41

34

10

0

3-2

4-9

35

10

0

4-4

4-2

36

10

0

4-8

MISCELLANEOUS EXAMPLES

293

Define the arithmetic mean, the median and the mode, and give a
sketch of a skew frequency distribution showing the approximate
position of each.

iState the chief advantages of the arithmetic mean as a form of
average, and find the arithmetic mean and the median for the wages
of agricultural labourers in the above table.

(30) p]xplain clearly the meaning of the term ' dispersion,' and
find the mean deviation from the median for wages in the same
table.

(31) Also, define standard deviation, and find the standard
deviation of these wages.

(32) Using the data in question (29), test graphically, with squared
paper, the correlation between average wages and percentages of the
population in receipt of poor law relief, stating your conclusions in
Avords.

(33) Construct a blank table, complete with headings and Unas, and
with due regard to spacing, in which could be inserted the numbers
of persons employed in six groups of industries, four grades of age at
three different periods.

(34) The following table gives for 780 weeks the call discount rate
and the ratio of reserves to deposits in New York. Calculate the
average discount rate for the various ratios of reserves to deposits,
and ex2:)ress the results graphically.

Call Discount Rates.

1-

2-

3-

4-

.5-

6-

r

s-

9 10

12-

15-

20-

25-

Totals.

Ratio of Reserves to Deposits.

217,-

237,-

257,-

277,-

29°/,-

317,-

337,-

357,-

37°/.-

397;-

417o-

437,-

457.-

"6
25
47
36
22
18
36
20
2
2

72

87

26

6

2

2

n

57
31
11

1

1

1

33

42

13

4

3

1
1
30
36
3
1

'2
27
16

■9
t
2

...

'3
6
1

1
"3

2
4

1

"i

>7

i

"i

...

1
0

"i

...

...

...
...

3

10

127

239

162

89

46

24

20

36

20

2

2

Totals, . 214

195

109

97

72

45

18

10

4

7

1

3

1

4 780

The heading 15- covers all rates of 15 and over but less than 16,

etc.

From the folloA\ing data find the equation of the regression line
giving the average discount rate for all ratios of reserves to deposits,

y\

2d4

STATISTICS

and plot the line on the same diagram. Is the use of the product
moment metliod of determining correlation justifiable in this case ?

Means.

Standard Deviations.

Correlation.

Call Discount Rates

Ratios of Reserves to Deposits

3-6
30-3

2-5
4-2

} -..

(35) As an illustration of the nature of definitions in statistics
explain fully the meaning of the statement : ' The total value of
exports (produce and manufactures of the United Kingdom) in 1918
was £498,473,065.'

MISCELLANEOUS EXAMPLES 295

PART II

(1) Give a short aocount of the cliief offieial publications, in
England, relating to statistics of one of the following subjects, with
especial reference to the source and the precise meaning of the data : —

(a) Vital statistics (births, deaths and marriages).
(6) Foreign trade,
(r) Agriculture.

(2) What do you understand by the words " frequency group ' ?

England and Wales, 1911

Ages . . . .10-15-20- 25- 35- 45- 55- 65-
All Occupied (Males 000s.) 246 1164 1146 2225 1815 1262 723 299
Coal Hewers (00s.) . . 63 338 538 1067 798 467 212 50

Comj)ute suitable a^erages and measures of dispersion for the
comparison of the age groups in this table and comment on the results.

(3) What means are available for testing the significance of
differences between statistical coefficients ? Test whether the
differences between the means and measures of dispersion for the
two series given in the previous question are significant.

(4) 5^=4-53 and .s.,=3-71 are the standard deviations of two groups,
.Tj, .i;^ . . . .T„, and y^, y.^ . . . y„. Sxy =^S32. n^lOOO.

Explain exactly the meaning of standard deviations. Calcu-
late the product-sum coefficient of correlation between the groups,
and state what it measures. Write down the probable error of the
coefficient and explain its meaning.

(5) Find the standard deviation of the differences between corre-
sponding values of two variates x and y.

(6) Set out in detail the method by which you would make graphic
comparisons of two such series of figures as Imports of Manufactures
and Unemployment.

(7) If the recorded births in a certain district may be in defect by
X per cent., and the estimated population in error Ijy iy per cent.,
find an approximate expression for the greatest possible error in the
birth-rate, x and y being assumed fairly small (say, not more than
5 per cent, or so).

(8) Given five thousand different figures — e.g. quotations of prices,
or measurements of human statures — how A\ould you (a) select five
hundred figures at random from that total, and (b) ascertain the
probability that the average of the five hundred selected figures does
not differ from the average of the five thousand bj^ more than any
assigned extent ?

U

296

(9) (a)

STATISTICS

>•' amber of Persons
per Tenement.

Number of Rooms per Tenement.

Total.

Approx.
Average
Numt)er

of
Rooms.

2

3

4

5

6

7

8

9

10
or more.

1
2
3
4
6
6
7
8

26
14
5
4
3
2

9

24

24

16

18

8

5

1

8
60
61
57
36
21
18
14

5
40
57
44
42
43
14
11

1
29
34
27
21

}?

10

4

1?

16
6
4
5

1

8
8
3
5
5
3
2

1
3
3
2
5
3
2
2

1
4

5
10

12
7
4

4

56
191
208
179
148
112
62
44

3-5
4-8
5-0
5-2
5-3
5-4
5-5
5-7

Totals, .
Average Number
of Persons,

M
2-07

105
3-56

275
394

256
4-24

152

4-24

55

3-78

35
414

21
4-62

47
481

1000
3-976

51

Standard deviations : persons 1-83, rooms 1-9 (approx.)-
(6) Show that the coefficient of correlation can be expressed in the
form

1

[~^{xy)-xyj,

where x, y are the averages of the observations referred to any origin.

(c) Calculate, by any method, a measurement of the correlation
between the number of rooms and number of persons per tenement
shown in (a).

(d) Calculate the third and fourth moments of the frequency
curve of persons ; determine the position of the mode and also
determine the skeAvness by any method knoTMi to you.

(10) The table given below gives the results of the measurement of
series of 959 Oxford Students and of 2348 convicts. Find what, if
any, differences between the statistical constants given are significant
and comment on the results.

Character.

Data.

Means.

Standard
Deviations.

Coefficients of Correlation
with Stature.

Head
length

f.Students
K'onvicts

196-05
192-44

6-23
6-39

•31
•26

Head

-breadth

/students
\Convicts

152-84
151-02

4-92
5-49

•14
•15

Head
-height

/Students
1^ Convicts

136-62
132-29

5-80
5-21

•28
•19

Stature

/Students
\ConA'icts

69-49
65-44

2-60
2-65

—

MISCELLANEOUS EXAMPLES 297

(11) Cive a short account of the nature of the information con-
tained in one of the following : Census of Production, 1907 ; Reports
on Wages in 1906 (the ' Wage Census ') ; Reports on Buildings and
Tenements (housing and overcrowding) in the Population Census,
1911.

(12) Outline a method by which the normal curve of error can be
obtained as the limit of {p-\-q)".

(13) Discuss (a) the best means of obtaining accurate statistics of
family expenditure, and (6) the best means of combining such data so
as to form a representative type.

(14) Define the following terms and give illustrations of their use :
interpolation, standard deviation, moment, skewness, logarithmic
scale, geometric mean, partial correlation, normal curve of error.

(15) In m trials an event has happened r times. How vi ould you
determine the probability that this result is consistent with the
hypothesis of random sampling from a universe in which the chance
of the event happening is a certain small quantity p ? Why cannot
the required probability be derived from a table of the normal curve
of error ?

(16) If m^, m.^ are the numbers of deaths occurring in a year among
Nj, Ng persons of two different occupations, the standard deviation
by which the significance of the observed difference in the death rates
per 1000 can be tested is given as

iooo^{^^^(^7^^^+^^^^^7^^^}.

Show how this formula is obtained and criticise it.

(17) Contrast the methods used in the construction of any two
current index numbers of wholesale prices. Under what conditions
is ' weighting ' important in index numbers 1

(18) Analyse in some detail the cases in which it may be assumed
(1) that a frequency distribution is normal, or (2) that the proba-
bility of errors in a measurement or observation exceeding various
amoimts is determinable by the normal table of probability.

(19) What methods are available for testing the ' goodness of fit '
of a mathematical curve to observations ?

(20) If z=a;i-(-a;.2+ • • • +-'^h, where the .r's are deviations from
the average of quantities selected at random and independently of
each other from a curve frequency whose standard deviation is o-,

show that the standard deviation of z is r-^, n being finite.

sin

Under what circumstances can it be shown that the curve of
frequency of z is normal ?

(21) What methods are available for classifying frequency curves
into types ? State briefly the mathematical concepts underlying the
Pearsonian classification of frequency curves.

298

STATISTICS

(22) Explain how the necessity for Sheppard's corrections of
moments arises.

If Wj is the second moment calculated from observations when all
are supposed to be grouped at the middle of grades whose breadth is

h, show that Wo+ -^ is the second moment if it is assumed that the
2 12

observations are evenly distributed through the grades.

(23) A sample containing 1000 is drawn at random from a large
universe and 300 are found to possess a certain attribute. Can you
infer anything as to the proportion in the universe that have this
attribute, or what further information is needed ?

(24) Discuss the effect on a weighted mean of errors in the
quantities or the weights.

(25) Write a brief note on the assumptions made in calculating the
probable error of a statistical quantity, such as the standard deviation
or the correlation coefficient.

(26) Calculate the average, second, third and fourth moments,
mean deviation, standard deviation and skewness of the frequencj^
groups of chest girths shown in the following table : —

Height and C

HE9T

Girth or

1126 Rkc

RUITS OF

L8Y

•-AR-S

OF A

QE.

Chest Girth
in Inches.

Height in Inches.

Totals.

tiO

til

62

63

64

65

66

67

OS

69

70

71

72

28

1

1

29

1

1

...

2

30

i

"i

3

1

1

2

9

31

2

!)

8

3

4

6

7

3

0

'2

i

47

32

8

18

24

29

36

12

16

5

8

2

1

159

33

6

U

21

30

42

22

36

17

8

9

i

3

i

207

34

6

16

1.5

43

52

43

21

40

28

12

1

3

280

35

6

15

25

32

32

2;i

29

18

16

14

2

i

219

36

4

3

6

11

22

18

18

18

19

U

2

127

37

1

3

1

"4

12

6

8

6

G

1

"i

49

38

•>

1

1

1

4

3

4

4

2

22

39

...

1

1

2

4

Totals, . .

23

68

91

140

180

143

144

121

97

71

30

14

4

1126

Averages of

Arrays,

32-7

33-6

33-5

34-2

34-1

34-7

34-7

350

351

35-5

36-3

35-4

34-5

34-51

Standard

Deviations

of Arrays, .

1-75

1-87

1-57

1-34

1-33

1-50

1-76

1-40

1-77

1-67

1-3

1-8

2-2

1-66

Average height, 65-6 ; standard deviation, 2-52.

Compare the relations between the quantities calculated with those
that are found in the normal curve of error.

(27) From the above data draw the regression line (chest girth on
height), and with the help of your drawing find an approximate value
of the coefficient of correlation between height and chest girth.

In normal distributions the standard deviation of an array is
o-j Jl—r^ where a-^ is the standard deviation of the arrays merged in
one group. Are the standard deviations shown in the table con-
sistent mth this formula ?

INDEX

[ Tht numheri^ refer to payei ; I and II refer to the two parti of the hooh.'\

Abscissa, I 68.

Abstract of Labour Statistics, I 19 ; II

177, 213, '283.
Advancing Dirt'erences, I 92.
Age Distribution of Criminals, I 04.
Anemone Neraorosa, II 227-30.
Arithmetif Mean (•'*ee Mean).
Array, I 109, 122-3, 127; II 251,

256-7, 262.
Assortative Mating, II 168.
Average, I 4, 9-11, 22-41, G3, 115,

125; II 136, 155-7, 173.

Bernoulli, II 149, 248.

Beveridge, I 15, 81.

Biassed and Unbiassed Errors, II
154-6, 170, 179.

Binomial, I 91 ; II 141-3, 146-7, 151,
181, 187-8, 190, 232-5, 240.

Biometrika, I 51, 103, 109, 114; II
134, 161-3, 165-7, 196. 212, 227,
230.

Birth Rate, I 7, 32, 82 ; II 234.

Board of Trade Index Number of
Prices, I 39.

Board of Trade Joxcrnal and Com-
mercial Gazette, II 282.

Booth, II 279.

Bowley, 137; II 170-3, 280.

Boyle's Law, I 75.

Brain-weight, I 103-4 ; II 166.

Bravais, 1 114.

Burnett-Hurst. II 173.

Calcdlation of Mean and Standard

Deviation, I 52-5.
Cattle and Grass-land, I 120-5.
Census, I 1, 14-15, 20, 40, 85, 117;

II 177, 280-2.
Central Statistical OfKce, II 279.
Centre of Gravity, I 60 ; II 269.
Chance, 14; II 134, 136.
Charlier, II 248.
Classification and Tabulation, I 4,

14-21.
Class-interval, I 21, 52, 103, 120-3;

II 195, 213, 218.

Coefficient of Correlation, 1 108, 110-12,

120, 129, 131 ; II 133, 152, 158,

162-3, 168-9, 253, 260, 278.
Coefficient of Variation, I 50-51 ; II

1.S3-4, 162-4, 167-8.
Compound Interest, I 7 ; II 263.
Condition of School Children, 117.
Constancy of Great Numbers, I 1.
Consumers' Surplus, I 97, 101.
Continuous Variation, I 26 ; II 182,

195-6, 232.
Co-ordinates, I 68.
Corrected Death Rate, I 32-5.
Correlation, I 4, 18, 76, 78, 82, 102-31 ;

II 151-3, 166-7, 253.
Correlation Ratio, I 112, 114; II 163.
Cost of Living, I 8, 29, 35-8, 125-31 ;

II 281-2.
Criminal Anthropometry, II 166.
Crude Death Rate, I 32-5.
Cuckoo's Eggs, II 161.
Cunynghame, I 95.
Curve Fitting, II 179-230.

Darwin, I 3.

Death Rate, I 7, 32-5. 82 ; II 234.

Deciles, I 49.

Decreasing Return, I 99-100.

De Moivre, II 149.

Descartes, I 68.

Dispersion, I 4, 42-51, 108; II 145,

162.
Distribution of Random Digits, II 157.
Dividend Sample, II 171.

Economist, I 9-11.

Edgeworth, I 3. 114; II 172, 230

248.
Elasticity of Wire, I 73.
Elderton, II 190, 203, 230.
Ellipse, I 69 ; II 252, 257-62.
Error, I 105.
Errors of Observation, I 4, 73 ; II

233.
Euleri II 149, 248.
Examination Marks, I 25, 27-8, 52-5.

66-7, 93 ; II 175-6, 207-212.

300

STATISTICS

Fechner, II 248.

Fermat, II 149.

Fitting of Curves, II 179-230.

Fitting with a Parabola, 1 8S-9.

Fluctuations of Sampling, I 47.

Flux, I 39.

Frequency, 1 12, 54, 60, 62, 102-3, 122-3,

127; II 135, 143, 150-3, 157, 172,

178-9, 184, 195-6, 200, 238-9.
Frequency Curve, I 60 ; II 178-230.
Frequency Distribution, 1 11-13, 52-67 ;

11 194, '267.
Frequency Polygon, II 180, 182, 184,

195.
Frequency Surface, II 249-62.
Funoti<»n, I 72, 87.

C4ALT0N, I 3, 49, 108-9, 114; II 248.

Gaiuma Function, II 214.

Gauss, II 248, 273.

Generalised Probability Curve, II 187.

Geometric Mean, I 38-9 ; II 264-5.

Goodness of Fit, II 180, 212, 219, 223-

224.
Graphs, I 57, 68-101.

Hain, I 3.

Halley, I 1.

Harmonic Mean, I 39 ; II 264-5.

Height of School Children, I 18.

Herschel, II 234.

Histogram, I 60; II 180, 219-20, 223,

225, 227, 232.
Homogeneity, II 165-6.
Household Budgets, I 35-6; II 155,

266.
Hyperbola, I 69.
Hj'pergeometiieal Expansion, II 187.

Tncreasikg Return, I 100.

Index Numbers, I 8-11, 29-31, 35-9, 76-
80 ; II 156.

Indictable Ofl'ences and Unemploy-
ment, I 82-3.

Infant Mortality, I 65, 117-19.

Infectious Disease Rate, I 56, 62 ; II
176, 219-27.

Inheritance, II 162, 167-8.

International Statistical Congress, I 2.

Interpolation, I 24, 48-9, 76, 85-94 ; II
212.

Journal of tht Royal Statistical Society,
I 8, 37 ; II 170, 230.

Kapteyn, II 248.
Knapp, I 3.

Labour Gazette, I 120 ; II 282.
Lagrange, II 149.

Lagrange's Interpolation Formula, I
94.

Laplace, II 149, 248.

Latter, II 161.

Least Squares, I 105-6 ; II 196, 271-3.

Lee, I 109 ; II 167.

Levis, I 3.

L'homme Moyen, I 2.

Limits for Correlation Coefficient, I

110.
Lipps, II 248.

Livelihood and Poverty, II 173,
Logarithmic Curve, I 87.
London Statistics, I 83, 117.

McAlister, II 248.

Macdonell, I 51 ; II 166.

Maclaurin, I 87 ; II 187.

Marriage Rate, I 32, 77-81, 115-17; II

234.
Marshall, I 95.
Mean— Arithmetic Mean, I 22, 30, 38-

41, 52-5, 60, 62-5, 104-13, 116-19, 122

5, 127-9; II 132-4, 144-5, 147-8, 153,

157-8, 160-4, 167-9, 193, 201, 204-5,

208, 217, 223, 226, 228, 238, 251.

256-8, 265-9, 273.
Mean Deviation, I 42-6, 50, 55, 64 -.

II 246, 270-1.
Mean Error, II 245.
Mean-square Deviation, I 46-7, 54 ;

II 211, 268-9.
Median, I 23-6, 36, 38-41. 43-6, 60-7.

87 ; II 162, 204, 238, 270.
Median Error, II 245.
Meteorological Observations, I 83-4.
Minority Report on the Poor Law, I 15.
Mode, I 26-9, 36, 38-41, 60-2, 64, 66-7 :

II 162, 186, 190-3, 204-5, 217, 223,

226, 228-9, 238.
Moments, I 123-4, 127; II 152-3, 163,

187, 194-205, 207-8, 213-16, 220-22,

225-6, 228, 267-9.
Monopoly, I 101.
Monthly Bulletin of Statistics, II 283.

Non-linear Regression, I 112, 114.
Non-measurable Characters, I 6 ; II

162.
Normal Curve of Error, 14; II 133,

184, 193, 206-12, 231-48. 251-2, 256.
Normal Demand, I 99.

Occupational Death Rate, I 34.
Ogive Curve, I 66.
Ordinate, I 68.
Overcrowding, I 20, 117-19.

Paish, I 9.

Parabola, I 69, 73. 88 9 : II 188. 197-9.

Pascal, II 149.

Pearl, I 103 ; II 166.

Pearl and Dunbar. I 51.

Pearl and Fuller, II 134.

INDEX

301

Pearson, I 3, 50-1, 60-1, 108-9, 112,
114; II 162, 165, 167, 184, 187, 191,
212, 230, 248, 255, 276.

Petty, I 1.

Philosophical Transactions, II 162.

Plotting of a Frequency Distnbution,

I 65-60.

Point of Inflexion, II 238, 245.

Poisson, II 248.

Poor Relief, I 88-92, 120.

Population according to Age, I 20.

Prediction, II 169.

Prices. I 8-11, 35-9, 76-81, 115-17, 125-

31 ; II 235, 265, 281-2.
Probability, I 3-4 ; II 132-49, 151, 181,

187, 231, 236-50.
Probabilitv Curve, 14; II 184, 187,

236.
Probability Integral, II 209, 239, 245,

275.
Probable Error, II 133-4, 145, 150-64,

245-6.
Probable Error of Mean, II 153-4.
Probable Error of Sum or Difference,

II 157-8.

Proceedings of London Mathematical

Societ7/, II 203.
Producers' Surplus, I 98, 100-1.
Product Daviation, I 106, 113, 116,

118-9, 123-5, 127-9.
Production Census, II 177, 282.
Progress of the Nation, I 88, 94.

QuADRATUKE Formulae, II 197, 209,

218 230
Quartile, 148-9, 64, 66-7, 87 ; II 245.
Quartile Deviation, I 48-50, 55, 64-5 ;

II 245-6.
Questionnaire, I 15.
Quetelet, I 1-3.

Random Errors, I 4.

Random Sampling {see Sampling).

Registrar-General, 13, 14, 33, 55; II

280, 283.
KegreBsion, Coefficient of. Line of,
. I 108-12, 117, 119, 125-6, 129-31;

I] 162, 253, 257, 260-1.
Rent, I 29-31, 101, 125-31 ; II 281.
Rowntree, II 173, 279.
Eoynl Society Transactions, II 230.

Sampling, Random Sampling, I 4,
47; 11 l.''.2-4, 145-77, 179, 182, 212,
236, 243.

Sauerbeck, I 8-10, 35.

Schuster, II 163.

Secular Trend, I 80.

Sheppard's Adjustment, I 104 ; II 203,

208, 213, 225.
Short-time Fluctuations, I 78, 80.
Significance, II 133, 159-68.
Simpson's Rule, II 219, 230.
Skew, Skewness, I 4, 49, 61-4; II

146, 180, 184, 186-7, 190-2,205,235,

247.
Sleep and Physical Condition, I 19.
Smooth Curve, I 60, 86 ; II 234.
Social Statistics, II 279.
Standard Deviation, I 46-7, 50-5, 63,

106-13, 116-19, 122-5, 127-9; II

133-4, 145-8, 151-4, 157-64, 166-8,

171-2, 176-7, 207-9, 240-6, 251,

256-8, 266-9.
Standard Error, II 134.
Standard Population, I 34.
Statical Analogues, II 269, 271.
Statist, I 9,
Stirling, II 248.
Successful Events, II 136-8. 140-8,

151-4. 169, 171, 174-5, 231, 240.
Successive Differences, I 91.
Supply and Demand Curves, I 95-101.
Siissmilch, I 1.
Symmetrical Distribution, I 61 ; II

180.

Table of Probable Errors, II 163.
Tawney, I 40 ; II 158.
Tax, I 100-101.
Temperature Records, I 65.
Todhunter, II 149, 248.

Unemployment, I 15, 81-3; II 213-19.

Variability, I 42-51, 61, 63, 107-108,
111 ; II 132-4. 162, 165-6, 247-8.

Variable, I 6-13, 72, 75, 87, 116, 122;
II 263-9.

Variation, I 6.

Variation in Arcella, 151.

Variation in the Earthworm, II 134.

Variation in Eupagurus Prideauxi,
II 163.

Variation in Plants, II 167.

Wages, I 8, 40-2, 125-31 ; II 173,

224, 227, 234, 281-3.
Weighting, I 30-8.
Weighted Mean, I 29-38, 263 4.

Yule, I 114; II 227.

Uniform with this Volume

BELL'S MATHEMATICAL SERIES

(Advanced Section)

General Editor: WILLIAM P. MILNE, M.A., D.Sc.
Professor of Mathematics, Leeds University

AN ELEMENTARY TREATISE ON DIFFERENTIAL
EQUATIONS AND THEIR APPLICATIONS

By H. T. H. PlAGGiO, M.A., D.Sc, Professor of Mathematics,

University College, Nottingham ; formerly Senior Scholar af St.

John's College, Cambridge. Third Edition. Demy 8vo. I2.r. net.

The earlier chapters contain a simple account of ordinary and partial

differential equations, while the later chapters are of a more advanced

cllaracter, and cover the course for the Cambridge Mathematical Tripos,

Part II., Schedule .'\, and the London 15. Sc. Honours. The number of

examples, both workctl and unworked, is very large, and all the answers

are given.

'With a skill as admirable as it is rare, the author has appreciated in every
part of the work the attainments and needs of the students for whom he
writes, and the result is one of the best mathematical text-books in the
language' — MatJieinatical Gaze/te.

A FIRST COURSE IN NOMOGRAPHY

By S. Brodki.skv, M.A., B.Sc, Ph.D., Professor of Applied Mathe-
matics at Leeds University. Demy 8vo. los. net.

Graphical methods of calculation are becoming of ever greater importance
in theoretical and industrial science, as well as in all branches of engineering
piMctice. Nomography is one of the most powerful of such methods, and
the object of this book is to explain what nomograms are, and how they can
be constructed and used. The book caters for both the practical man who
wishes to learn the art of making and using nomograms, and the student
who desires to understand the underlying principles. It is illustrated by
sixty-four figures, most of which are actual nomograms, their construction
being analysed in the text. In addition, there are numerous exercises
illustrative of the principles and methods.

'A good introductory treatise . . . calculated to appeal to the student who
desires to make early practical use of the knowledge he acquires.'

T/ic Mfchanicnl World.

ELEMENTARY VECTOR ANALYSIS

With application to Geometry and Ph) sics. By C. E. Weatherburn,
M.A., D.Sc, Professor of Mathematics, Canterbury University
College, Christchurch, N.Z. Demy 8vo. I2J-. net.

This book provides a simple exposition of Elementary \'ector Analysis,
and shows how it may be employed with advantage in Geometry and Mathe-
matics. The use of \'ector Analysis in the former is abundantly illustrated
by the treatment of the straight line, the plane, the sphere and the twisted
curve, which are dealt with as fully as in most elementary books. In
Mechanics the author has explained and proved all the important elementary
principles.

LONDON: G. BELL AND SONS, LTD,

BELL'S MATHEMATICAL SBRiES— Co ntd.
(Advanced Section)

ADVANCED VECTOR ANALYSIS

With Application to Mathematical Physics. By C. E. Weather-
KURN, D.Sc, Professor of Mathematics, Canterbury University
College, Christchurch, N.Z. Demy 8vo. I5.f. net.

'The author has already published in this series an Elementary Vector
Anaivsis. In this companion volume he deals with the higher part of the
subject and the applications of the theory, including a chapter on the Equa-
tioins of Maxwell and Lorentz and the Lorentz-Einstein transformation.'—
Times.

THE ELEMENTS OF NON-EUCLIDEAN GEOMETRY

By D. M. Y. SOMMERVILLE, M.A., D.Sc, Professor of Mathematics
Victoria University College, Wellington, N.Z. ^s. 6d. net.

'An excellent text-book for all students of Geometry.' — Nature,
'A useful and stimulating book.'-- Mathematical Gazette.

ANALYTICAL CONICS

By D. M. Y. SOMMERViLLE, M.A.. D.Sc. Demy 8vo. 15.C. net.

In this book the elementary properties of the Conies are first of all dealt
with, and thereafter the higher portions of the subject, such as Conies
referred to any axes, Homogeneous Co-ordinates, Invariant and Covariant
]iroperties of Conies, etc. There are abundant collections of examples in
all the subjects treated.

A Professor of Mathematics writes: 'I find it a work entirely praise-
worthy ; and indeed of such excellency as one would expect from the pen
of the author of The Elements of Non-Euclidean Geometry. The variety of
topics treated is more extensive than that to be found in most existing text-
books on the subject. In certain instances the treatment is refreshingly
novel, and in all cases the presentation is concise and lucid.'

A TEXT-BOOK OF GEOMETRICAL OPTICS

By A. S. Ramsey, M.A., President of Magdalene College, Cambridge.
New and Revised Edition. Demy 8vo. 8^. bd.

This is a revised edition of the work published in 1914. It contains
chapters on Reflection and Refraction, Thin and Thick Lenses, and Com-
binations of Lenses, Dispersion and Achromatism, Illumination, The Eye
and Vision, Optical Instruments, and a chapter of Miscellaneous Theorems,
together with upwards of three hundred examples taken from University an
College Examination Papers. The range covered is somewhat wider tha.
that recjuired for Part I. of the Mathematical Tripos.

LONDON: G. BELL AND SO N S, LTD.
York House, Portugal St., W.C. 2.

A

5^^